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PARAMETRIC CONFIDENCE BANDS ON

CUMUIATIVE DISTRIBUTI<E FUNCTICES

by

Paul Benjamin Kanofsky

Institute of Statistics Mimeo Series No. 444

July 1965

This research was· supported in part by the Public HealthService Grant Nos. 2G-38, GM-38, and 64-509; in part by theNational Institutes of Health Grant Nos. PH 43-64-586 andGM 12868-01 ~ and 1.n part by the Air Force Office of ScientificResearch Grant No. AF-AFOSR-760-65.

DEPARTMENT OF STATISTICS

UNIVERSITY OF NORTH CAR0LmA

Chapel Hill, N. C.

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ABSTRACT

KANOFSKY, PAUL BENJAMIN. Parametric Confidence Bands on Cumulative

Distribution Functions. (Under the direction of NORMAN L. JOHNSON).

This dissertation is basically an investigation of how, assum-

ing a certain distributional form for the population being studied,

a chance selected region in the plane may be found such that the graph

of the c.d.f. (cumulative distribution function) of this population

lies in the region with a certain preassigned probability. If, in

addition, the intersection of the region with any vertical line is a

straight line segment, then the region is called a confidence band.

The problem. here is a particular case of the more general prob-

lem. of finding s:i.mu1.taneous confidence sets for a set of parametric

:functions. Some theory for the general problem is developed in

Section 1. A crucial concept defined there is that of a matting.

This concept may be explained as follows. For any particular value

assumed by a parameter point ~ each parametric :function of interest

also assumes a certain fixed value. Thus as the parameter point ~

varies over a certain region m, any parametric :function also varies

over a particular corresponding set. The collection of such sets

generated, if we consider all parametric :functions of interest, may

be referred to as M(m), the matting based on m. If the set over which

each parametric function varies is an interval, M(m) is called a band.

If e is a particular value of the parameter point ~ then the-0

set of values assumed by the parametric :functions of interest for

e = e may be referred to as G( e). The statement, "G( e ) is in- -0 -0 -0

M(m)", means that the value which each parametric fUnction assumes

for e = e is in the set over which the function varies for parameter- -0

value of that parametric function. Any given iso-g surface is the

The major result of the dissertation is a statement of condi-

side of this iso-g surface is the set of all parameter points that

surface is the set of all parameter points that yield value s of this

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The lower-

yield values of this particular parametric function smaller than or

r:hown that the iso-G surfaces with respect to values of the c. d. f.

are often straight lines in the parameter space. Thus, since a

of normal, exponential, and uniform random variables. When both the

In Section 2, confidence bands are constructed for the c.d.f. 's

caual to w. It is shown that the intersection of sides of connectedo

i~o-g surfaces is a g-wise eXhaustive region.

location and scale parameters for a population are unknown, it is

set of all parameter points that yield a particular value w , say,o

for a particular parametric function. The upper side of this iso-g

ment involves the concept of an iso-g surface. An iso-g surface is

characterized by a particular parametric function and a partiCUlar

tions which ensure that a region fA is g-wise exhaustive. This state-

that if!R is a chance selected region and also g-wise exhaustive,

then the probability that mcontains any particular parameter point

e is the same as the probability that M(!R) contains G( e ).-0 -0

points Q in m. It is clear that whenever e is in m, G( e ) is in M(9t).-0 -0

However it is not true for general m, that when e is not in!R, G( e )-0 -0

is not in M(m). Thus if!R is such that for e not in !R,G( e ) is not in-0 -0

M(!R), mis called g-wise eXhaustive, where g is a general denotation

for any of the paraJOOtric functions of interest. It is iDmediate

particular parametric function larger than or equal to w •o

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connected intersection of sides of straight lines is a Calvex figure,

it is found that certain convex figures are g-wise exhaustive.

Section 3 deals with criteria by which confidence bands on

c.d.f. 's can be eValuated. Some criteria treated are expected width

of the confidence band at a preselected value of the random variable

being studied, expected maximum width of the confidence band over all

values of the random variable being studied, maximum expected width

of the confidence band, 8Zld expected area of the confidence band.

Some of the parametric bands are compared with respect to certain of

the criteria to the Kolmogorov-Smirnov band, which is a non-parametric

band. As one would guess, the parametric bands perform better on all

criteria examined than the Kolmogorov-Smirnov band. For example, let

us consider the ratio of the expected maximum. width of a certain

easily constructed parametric band to the expected maximum lColmogorov­

Smirnov band width when a sample of size 91 is taken from a normal

population. (91 appears to be large enough that the ratio is closely

approaching an asymptotic value at this sample size.) This ratio is

about .78 if both the mean and variance are unknown, about •58 if only

the mean is unknown, and about .25 if only the variance is unknown.

In Section 4, consideration is given to putting sinmltaneous

confidence intervals on the values of the true c.d.f. overs. pre­

selected interval of values of a random variable, known to be normal,

but with unknown ~an and variance. Also treated here is the converse

problem of putting simultaneous confidence intervals on values of the

percentiles of this random variable over a preselected interval on

the ordinate scale. Finally, the construction of confidence bands

for multivariate c.d.f. 's receives some attention.

Born:

Degrees:

Employment:

iiBIOGRAPHY

April 30, 1936, Philadelphia, Pennnylvania.

B.A. in Psychology, 1956, Temple University,Philadelphia, Pennsylvania.

M.A. in Mathematics, 1960, Temple University,Philadelphia, Pennsylvania.

Statistician, Department of Public Health,City of Philadelphia, September 1958 to June 1960.

Fellow in Department of Biostatistics under .~

Public Health Service Training Grant, Universityof North Carolina, Chapel Hill, September 1960to May 1961 and September 1961 to December 196U.

Statistician, National Cancer Institute, Bethesda,Maryland, June 1961 to September 1961.

Research Associate, Department of Biostatistics,University of North Carolina, Che;pe1 Hill,.January 1965 to May 1965.

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iiiACKNOWLEDGMENTS

I would like to thank Dr. N. L. Johnson, my adviser, for his

tolerant understanding of my wild rambling and groping, and his

forceful aid in nailing down elusive concepts and proofs. Perhaps

most important has been his cheering encouragement throughout the not

always pleasant process called writing a thesis.

I would also like to thank Dr. J. E. Grizzle, under whose

tutelage I acquired the background with which to frame and develop

my thesis topic.

Of course, my patron saint during my career at North Carolina

has been Dr. B. G. Greenberg, who has helped me to meet and surmount

numerous variegated problems. His advice ani mere presence on the

scene have always been most reassuring and comforting.

It is only fair that I acknOWledge my :position as a ward of

the Federal Govermnent during the great portion of my time as a stu­

dent of statistics. To be specific, the United States Public Health

Service has seen fit to provide a living allowance for me over

several years, and the United States Air Force has financed the re­

production of this thesis.

The typing of the thesis has been done by Mrs. Dawn Lykken,

who has admirably endured and persevered throUghout the emendaticns

of manuscript attendant upon my ever broadening vis ion.

At the risk of being maudlin, I gratefully own to support and

stimulation from fellow students, Elmer Hall and Jay Glasser, who

have never failed to calm my irrational fears and prod me with con­

structive criticism and challenge•

Finally, I thank my parents who are ultimately responsible,

in many more ways than one, for any merit my 'WOrk may possess.

iv

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2. CONSTRUCTION OF CONFIDENCE BANDS FOR UNIVARIATECUMUIATIVE DISTRIBUTION FUNCTIONS • • • • • • • • • • • • 23

2.1 Theory Useful for Putting Confidence Bandson the Cumulative Distribution Functionsof Transformed Random Variables. • • • • · • • · • 23

2.2 Linear Transformations • • • • · • • • • • · • • • • 292.3 Explanation of a General Procedure for .

Constructing Bands • • • • • • • • • • • • • • • • 342.4 Normal Distribution. • • • • • • • • • • • • • • • · 36

2.4.1 Band Construction for a Normal Distributionwith Unknown Mean and Known StandardDeviation • • • · • • • • • • • • • • • • 36

2.4.2 Band Construction for a Normal Distributionwith Known Mean and Unknown StandardDeviation. • • • • · • • • • • • • • · • 37

2.4.3 Band Construction for a Normal Distributionwith Unknown Mean and Unknown StandardDeviation • • • • • • • • • • • • · • • · 39

2.4.4 Confidence Regions for the Parameters ofthe Normal Distribution and TheirSuitability for Band Construction • • • • 43

2.5 Exponential Distribution • · • • • • • • • • • • • • 55

2.5.1 Band Construction for an ExponentialDistribution with Unknown Initial Pointand Known Standard Deviation. • • • • • •

2.5.2 Band Construction for an ExponentialDistribution with Known Initial Pointand Unknown Standard Deviation••••••

2.5.3 Band Construction for an ExponentialDistribution with Unknown Initial Pointand Unknown Standard Deviation. • • • • •

Uniform Distribution • • • • • • • • • • • •

SOME THEORY PERTAINING TO SIKJLTANEOUSCONFIDENCE SETS • • • • • • • • • • • 4

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55

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TABLE OF CONTENTS

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LIST OF TABLES.

LIST OF FIGURES.

INTRODUCTION. •

1.

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TABLE OF CONTENTS (continued)

3.3'.2.1 Unknown Mean, Known Standard Deviation.3.3'.2.2 Known Mean, Unknown Standard Deviation.3.3.2.3 Unknown Mean, unknown' Standard

Deviation. • • • • • • • • • • • • • •

3.3.3.1 Unknown Initial Point, Known StandardDeviation. • • • • • • • • • • • • • •

3.3.3.2 Known Initial Point, Unknown StandardDeviation. • • • • • • • • • • • • • •

3.3.3.3 Unknown Initial Point, Unknown Standard.Deviation. • • • • • • • • • • • • • •

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90

66

90

lO?

102

107

10n

107

111

101)

111

112

• •

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.. . . .· .

· . . .

• •

. .• • • ••• • •

. . . .. . . . .

Preliminary Remarks. • • • • • •Maximum Expected Width of Bands'

for the Normal Distribution. •

Maximum Expected Width of Bands' forthe Exponential Distribution • • •

2.6.1 Band Construction for a Uniform Distribu­tion with Unknown Mean and Known Range • .

2.6.2 Band Construction for a Uniform Distribu­tion with Known Mean and Unknown Ranr:e • •

2.6.3 Band Construction for a Uniform Distribu­tion with Unknown Mean and Unknown Range •

2.6.4 Confidence Regions for the Parameters ofthe Uniform Distribution and TheirSuitability for Band Construction. • •••

3.2.1 General Theory • • • • • • • • • • • • • • •3.2.2 The Maximum Absolute Dif~erence' Between

Two Normal c.d.f. 'so •••••••••••3.2.3 The Maximum Absolute Difference Between

Two E)~onential c.d.f. 's •••••••••3.2.4 The.Maximum Absolute Difference Between

1'\'10 Uniform c. d. f. 's • • • • • • • • • • •3.2.5 The Distribution of ~IF(X;~)-F(x;~)I

in Certain Cases • • • • • • • • • • •

Introduction. • • • • • • • • • • • • • • • •The Maximum Absolute Difference Between

Tw'o c. d. f. 's. • • • • • • • • • • • • • . . . .

Expected Ma.x:i.nn.un Band Width • •

CRITERIA FOR CONFIDENCE BANDS. • •3.

7.' Computations for Tables •••••••••••••• •

7.1 Theoretical Supplement to Section 1. • • • • • • • •7.2 Theoretical Supplement to Section 2 • • • • • • • • •

116

116116116

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125133

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166169

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Numerical Results on Expected Maximum BandWidth for the Normal Distribution. • • • •

Invariant Properties of Certain ConfidenceBands. • • • • • • • • • • • • • • • • •

3.3'.4.1 Unknown Mean, Known Range••••••3.3'.4.2 Known Mean, Unknown Range••••••3.3.4.3 Unknown Mean, Unknown Range•••••

4.1.1 Preliminary Remarks•••••••••••••4.1.2 A Confidence Band for a Normal ·c.d.f

on a Selected Interval • • • • • • • • • •4.1.3 A Confidence Band for Percentiles of a

Normal Popllation in a Selected Interval •

7.2.1 Convex Figures • • • • • • • • • • •7.2.2 Theorems Used in SUb-section 2.6.4 •

7.'.1 Computations for Table 3.1 '. • • • • • • • •7.'.2 Computations for Table '.2 ••••••.•.

Expected Width of Confidence Bands' atSelected Points • • • • • • • • • • • '. •

Other Criteria. • • • • • • • • • • • • • •

Confidence Bands on Selected CumulativeDistribution Function VaJ.ues and Percentilesof a Normal'Random Variable ••••••••

Confidence Bands on Multivariate CumulativeDistribution Functions. • • • • • • • • • •

4.1

4.2

EXPLORATION OF FURTHER TOPICS. • •

LIST OF REFERENCES •

APPENDIX ••••••

SUGGESTIONS FOR FUTURE RESEARCH. •

~IE OF CONTENTS (continued)

3.3.4 Maximum Expected Width of Bandsfor the Uniform Distribution ••

4.

6.

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LIST OF TABLESviii

3.2 Expected width at selected points (E(W(z;BK_S» andE(W(z;BN»)of .95 confidence bands for a standardnormal distribution on a sample of size 21~ • • • • • • • 132

7.1 Procedure far computing ordinates in application ofSimpson's Rule -to evaluatioo of E(W(zo;~»' forPo = .25, .15, .10, .04, and .01 • • • • • • • • • • •• 185

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124. . .. .3.1 Expected ma.x:i.mum width (E(WM» of .95 con:f'idencebands for a normal distribution. • • • • • • •

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LIST OF FIGURES

2.1 C.d.f.-wise exhaustive confidence region on parameters(IJ.: ,0' ) of a normal distribution • • • • • • 0 0 0 •o 0

2.2 C.d.f.-wise exhaustive confidence region on parameters(7

0,0'd) of an exponential distribution • • • 0 0 0 •

2.3 C.d.f.-wise exhaustive confidence region on parameters(~ ,w ) of a uniform distribution•• 0 ••• 0 0 0 0o 0

4.1 Cod. f.-wise exhaustive confidence region on parameters(~ ,0' ) of a normal distribution for values in thein£er~al [XA, ~] • • 0 0 0 • 0 0 0 0 0 0 • 0 0 0 0 0

4.2 Confidence region on parameters (~lO'~O'O'O) of acircular normal distribution • 0 0 • 0 0 0 0 • • 0 0

o 0

o 0

o 0

o 0

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INTRODUCTICE

Wald and Wolfowitz [1939] treated the following problem.

Suppose X is a r.v. (random variable) with a continuous c.d.f.

(cumulative distribution function). In the absence of all other

knowledge about X, find an interval for each popllation value F(x),

- co < X < co, such that the probability that all intervals, (i.e.,

for all values of x) contain the true values of F(x) is some pre­

assigned value. T1>at is, find a set of simultaneous confidence

intervals for the set of population values, F(x), - co < X < co. Let

us refer to the union of these simultaneous confidence intervals as

a confidence band on the c.d.f, F(x). Wald and Wolfowitz showed that

a variety of such non-parametric confidence bands exist. Kolmogorov

and Smirnov obtained results which facilitate construction of one

such band. This band is based on the D -statistic, max!F(x)-SN(x) I,n x

where SN(x) is the sample distribution function far a sample of size

n [Kendall and Stuart, Vol. 2, 1961].

In this thesis we will investigate how one may put a confi­

dence band on a c.d.f. if its distributional form is known, e.g.

we may know that the c. d. f. we are dealing with is that of a normal

or exponential r.v. In such a case, the only additional items of

information we need, to completely specify the c. d. f., are the

values of the parameters involved in the form of the c.d.f. If our

knowledge of the distributional form is utilized in constructing a

confidence band for the c.d.f., we will say the confidence band

produced is of a parametric nature in contradistinction to the

non-parametric bands discussed by Wald and Wolfowitz.

We will refer to the confidence band m a c.d.f. based on the

Kolmogorov-Smirnov D -statistic as a K-S band. Owing to the non-n

availability of tables for any other non-parametric bands, we will

confine ourselves to comparing parametric bands with K-S bands.

The problem of cmstructing parametric confidence bands on

c.d.f. 's is subswned under the general problem of simultaneous con­

fidence interval estimation of parametric functions as stated by

Roy and Bose [1953]. We will discuss a method of constructing para.­

metric confidence bands in keeping with the approach to such

problems proposed in their paper. However this method will not be

developed or followed to any great extent.

/Instead some remarks by Scheffe [1959, p. 82] have been

generalized. He points out how one could study the problem of

multiple comparisons of linear functions of the parameters in the

general linear :nx>del by starting with a "convex confidence set" for

the parameters, rather than with the linear functions of the param­

eters. In the first chapter of this thesis, it is shown how one

can study multiple comparisons of any parametric functions starting

from a confidence set (for the parameters involved) that possesses

a certain generalization of the property of convexity mentioned by

/Scheffel

Assuming a normal POPUlation with both mean and variance un­

known, Johnson and Welch [1939] showed how one cculd form a confi­

dence interval for any specified percentile. They also showed how

one could form a confidence interval in this case for the value of

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the c.d.f. at any specified point. However they did not consider

simultaneous confidence intervals on c.d.f. 's and percentiles. In

Section 4, their results are used to form confidence bands on a

normal c.d.f., that 'Would appear to be particuJ.arly good for a pre­

chosen x-interval or for a pre-chosen range of percentiles.

:3

1. SOME THEORY PERTADiING TO

SIMULTANEOUS CONFIDENCE SETS

A major mtivation of this thesis is to ascertain to what extent

confidence bands on c.d.f. 's can be improved, if one makes more assump­

tions about the c. d. f. than are made in the non-parametric case.

One way in which additional information about the c.d.f. can be

used to improve a confidence band is the following. Put a K-S confidence

band on a c.d.f. Then make the added assumption the c.d.f. is one of a

family of c.d.f. 's, for instance normal, with unknown mean and variance.

On the basis of this assumption, the original K-S band can be "whittled

down" or reduced to the set of all normal ogives that fall entirely with­

in the original K-S band. The general theory in this section will be

introduced as a formalization of this type of approach to the SUbject.

In the overall development of this theory, except for remarks

interspersed between theorems and definitions, no explicit reference

vlill be made to c.d.f. ·s. However the theory does apply to c.d.f. 's,

a.s later sectionswill make clear. So as not to in;>ede the flow of the

argument some of the standard mathematical and statistical terminology

and theory used will not be reviewed in the body of the thesis, but will

be gathered together for reference in an appen~ An asterisk follow­

ing a term will indicate that term is discussed in the appendix.

Let w = g(v; 'Il) be a real valued function of v, defined over {v}.

Let g(v; 'Il) be one of a family {g} of sueh functions with index

set ('Il).

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5

Note here that {v} and {T}} are simply abstract sets. However

g{v;T} ) can be taken to be F{~;~) = y, the c.d.f. of ~, a vector of

random variables, where ~ is a parameter vector.

Let {G} be the set of graphs corresponding to {g}. That is,

G{T}) is the set of points {(v,w) Ive{v}, w = g(v;T})} in the product

set (v) x E , where E is the set of all reaJ. numbers. {G} is also

indexed by {T}} •

Let S be a chance selected subset of {v}x E •

Let Tla be the true vaJ.ue of T}.

Suppose we know Pr(G( T}o) C S). That is, S is a confidence

region for G{T}o). It may be possible to reduce S by using all the

information available to us. This can be seen if we note that

Therefore PR(G( T} ) C S) = Pr(G{ T} ) C U (G IG C S». Thus U(G IG C S)o 0

is a confidence region for G(T}) with the same confidence coefficiento

as S. If U(G IG C S) is a proper subset of S, we say we have reduced

the original confidence region S to a smaller confidence region

u{GIG C S).

It is of interest to find conditions which guarantee that if S

is a band, U{G IG C S) is also a band. To do this we need a precise

definition of a band and of severaJ. other concepts.

B, a subset of (v}x E, is a band iff for every v in {v) ,- .

{w I(v, w)eB) is an intervaJ...

The uPper side t Su( T}), of G{ T)l in (G J., is defined as

Su(T}) = ((v,w) I g(v;T}) 2= w; (v,w)e {v} x E) •

6

The lower side, SL(")' of G(!l) in {G}, is defined as

Some remarks on notation are appropriate at this point. We use

*the conventional symbol [a,b] to designate the closed interval from a

to b where a and b are real numbers. However an extended mean-

ing will adhere to this symbol if a and b are not real numbers.

Thus we say the subset, [G("l)' G(~)], of {v} x E, is defined as

[G("l)' G(~)] = (U(SU(~) I i=1,2»n(u(sL("i) I i=1,2».

In the next definition we adopt a form of expression which will

be used elsewhere as well. That is, if Q is a set, then ql'~€ Q

will mean ql € Q and ~€ Q ~ ~,~c Q will mean ~ c Q and ~ c Q •

Now (G) will be termed comprehensive iff

G("l)' G(~)€ (G) -+ [G(~), G(~)] = u(GIG€(G}o)

where (G }o c (G) •

We proceed to denxmstrate some results culminating in the theorem

that if B is a band, and (G) is comprehensive, then U(G IG c B) is

a band.

Theorem 1.1

(WI(VO,W)E [G("l),G(~)]) =

= [min(g(vo;~),g(vo;~»' max (g(vo;~),g(vo;1'12»].

Proof: Let WoE (w I(vo' w) € [G( ~),G(~)]} •

By the definition of [G(~),G(~)], (vo'wo)€ U (SU(~) I i=1,2).

That is, g{v ;"1) <w, or g{v ;11..) <w •o -0 o~-o

Hence min (g{v ."1)' g(v ;11..» < w •o o·~ - 0

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7

Also by the definition of [G(Tll),G(~)],(vo'wo)€U(St(Tli ) I i=1,2).

Hence w < ma.x(g(v ;n..), g(v,n...».0- 0 '.L 0 ~

Thus min(g(v ;Tl1),g(v ;n...» < w < ma.x(g(v ;Tl1

), g(v ;n_».o 0 ~ - 0- 0 0 c

Now let w €[min(g(v ;n..),g(v ;n...», ma.x(g(v ;Tl1),g(v ;n...»] •o 0 ~ 0 ~ 0 0 c

w' < ma.x{g(v ;Tl1 ), g(v ;n...»~ (v ,w )€ U(St(Tli ) I i=1,2) •0- 0 0 ~ 0 0

min(g(v ;Tl1), g(v ;n...» <w ~(v,w )€U(Su{n.)I i=1,2) •o o~ -0 00 '1.

Thus by the definition of [G(~),G(~)], wodwl(vo,w)€[G(,\),G(~)]}•

G-Ql'ollary 1.2. (,G(Tli),G(~)] is a band.

Proof: g(v; Tl1 ) and g(v;~) are defined for every element of {v}.

Hence for every v in {v}, by Theorem 1.1, (wl(v,w)€[G(~),G(~)]}

is an interval. Thus [G(~),G(~)] is a band, by definition.

Lemma. 1.3 If B is a band and G(Tl1 ) and G(~) are any two

graphs in B, then [G(Tll),G(~)]c B.

Proof: Assume (vo,wo)€ [G( ~),G( ~)].

Then Wo € [min(g(vo ; ~),g(vo;~»' ma.x(g(vo ;~); g(vo;~»]' which

*is a closed interval with end points in {w I(v , w) € B), which is alsoo

an interval by the definition of a band. Now if a closed interval W1

has its end points in another closed interval W2' then W1 c W2 •

Hence w e{w I(v , w) € B}. That is, (w, v ) € B.o 0 0 0

Theorem 1.4 If T' is a band, and (G) is comprehensive, then

U(G IG c B) is a band.

Proof: We want to show (wl(vo,w)e: u(GIG c: B)} is an interval,

where v0 €(v). Let wl and w2' wl < w2' be any two points in this

set where (Vo,wl)e: G(~) c: B and (vo'w2 )€ G(~) c: B.

Thus wl = min(g(vo,T)l)' g(vo;~» and w2 = IIBX(g(VO'~)' g(vo;~».

Now if we can show that

[wl'w2 ] ,c: (wl(vo'w)€ u(GIG c: B)} ,

the proof is complete •

As shown in Theorem 1.1, [wl'w2 ] = (wl(vo,w)€[G(~),G(~)]} •

Since (G) is comprehensive, [G(Tl.l),G(~)] = u(GIG€{G}o c: (G}).

Thus since G(~) and G(~) are in B, U(G IG€(G}o) c: B •

Hence if w €(wl(v ,w)€[G(n. ),G(n-)]} , w €(wl(v ,w)e: u(GIG C B)}.o 0 ~ ~ 0 0

In the second section, some connnonly used c. d. f. families will

be shown to be comprehensive.

Now if we identify 1) as a lxp vector of parameters, !, another

technique for putting a confidence band on a c.d.f. graph can be

suggested. Consider the random variable mp IF(!.;~) - F(292o'1 = d(say},

where ~ is some appropriately chosen est~tor of Jb- Of course the

distribution of d will depend on the family of c.d.f.'s involved

and on i. For the time being we will avoid specifying d :fUrther.

Note that if the distribution of d is known, we can put simul­

taneous confidence intervals on the set of functions, F(!.;~o)'

(for - 00 < x < (0) of the parameter e , which are together equivalent- - - -0

to a confidence band on the graph of F(x; e )in the following wa:y.--0

Let ~-a be the point above which d falls with probability

ex. Then

II-.IIIIII

_IIIIIIII

_II

which is a band.

taneous interval. estimation. We sh8J.1. discuss it with reference to

9

Now we will explore the relation between confidence regions for

The resultingS0100 commonly used c.d.f. families, in Section 3.

110 and for G( 110).

A confidence region, S, for G(~) always yields the confidenceo

region, (~IG(~) c S} , for 110. As 110 €(~ IG(~) c S}~ G( 110) c S ,

the confidence coefficient attached to S applies to the confidence

bands on the graphs involved can be reduced as explained previously.

Roy and Bose [1953] use this approach to the problem of simul.-

A· AT IF(!;~)-F(!;!W l.:s ~-a ~ IF(!; e)-F(!;~) l.:s '\-a ' all real b

A A~F(!;~) - ~-a .:s F(!;.2;).:s F(!;~) + '\-a' all real! ,

~ the graph of F(!;§dC {(!;y) I -~ <! < ~ ,

F(!;~ - '1.-a.:s y .:s F(!;~ + '\-a} ,

dence band on a linear regression line are to be found in Working andI

Hotelling [1929], and Scheffe [1959].

region for ~ derived from it.o

Here, however, our problem is to find confidence regions for

G(~ ), not ~. As confidence regions for ~ are often available,000

we are led to consider confidence regions for G(~) based on confi-. 0

dence regions for ~o. The general idea is to transfer the points in

a confidence region for 110 to the set (v) x E in the forms of the

corresponding graphs and then to use the set formed by the union of

these graphs as a confidence region for G( ~o).

PreTious uses of the 1OOthod, in connection with putting a confi-

II.eIIIIIIIeIIIIIIII_I

10

To begin with, we state some terminology and notation.

Let ~ be a subset of {~}. Define M(~) as U(G(~) I~€ ~).

We refer to M(!R) as the matting based on ~ •

Now if ~ is a confidence region for ~ , we would likeo

Pre ~o€ m) = Pr(G( ~o) c M(~». Since, by the definition of M(~),

~ € m ... G(~ ) c M(!R) , we have Pr(~ €~) < Pr(G(~ ) c M(m». Thuso 0 0 - 0

if G('I'}) c M(m) ... ~ € ~ , then Pr(G(~) C M(!R» = Pr(~ € m). Weo 0 0 0

will say that m is g-wise eXhaustive iff G( ~o) c M(m) ... ~o € ~ •

The terminology here is motivated by noting that

So we have the following theorem:

It is helpful to define an iso-g surface" C(v,w), as

(7) Ig(v;~) = w; ~€{~}}, where v and w are fixed. That is, an

iso-g surface is the set of points in {~} for which g(v;~) is

constant at w. Call (v,w) the ::g,oint of confluence corresponding to

the iso-g surface C(v,w). C(v,w) consists of all ~ € {~} , whose

corresponding graphs pass through the fixed point (v,w) in (v) x E •

Lemma 1.6 G( ~l) c M(~), iff every iso-g surface through Tl1

intersects ~.

Proof: To prove sufficiency, we note that if (vl,wl ) is any

point on G(~l)' then ~l€ C(vl,Wl ) and by hypothesis C(v1,w1 )

intersects !R at ~ , say. Thus by the definition of an iso-g sur­

face, (vl

, wl ) € G(~) c M(~~) • Thus G(~) c M(m) •

II-.IIIIII

_IIIIIIII

_II

II.eIIIIIIIeIIIIIIII_I

11

To prove necessity, consider any iso-g surface Cl through 'Ill •

Let (vl,wl ) be the point of confluence corresponding to Cl

• Now

(vl'wl )€ G('Ill ) by the definition of Cl • But G('Ill)C M(m). Hence

(vl'Wl )€ G(~), where ~ € m. Thus ~€ Cl • That is, Cl inter­

sects m at ~ •

Theorem 1.7 m is g-wise exhaustive, iff every point in ('Il) not in

m has at least one iso-g surface through it, no point of which is in m.Proof: First we will denx>nstrate the sufficiency of the condi-

tion. If 'Ill" m , G( 'Ill) '!- M(m), by the previous lemma. Hence m

is g-wise exhaustive, by definition.

Necessity follows if we note that the conditions- m being g-

wise exhaustive and '1)1" m - imply G('Ill ) cf. M(m), by definition of

g-vdse exhaustiveness. However if all iso-g surfaces through 'Ill

intersect m, then G( 'Ill) C M(lR) by the previous lennna. In the

light of this contradiction, at least one iso-g surface through M(m)

must not intersect m.

In addition to wanting a confidence region for G( Tlc) to have an

exact confidence coefficient attached to it, we would also like it

to be a band. Thus we make the following definition.

m is a band producing region iff M(m) is a band.

~eorem 1.8 m is band producing iff

Proof: First assume-

J2

For the second part of the theorem, assume m is band producing,

*Apropos of the next result, we remark that a topological space,

II-.IIIIII

_IIIIIIII

_II

=w •oin.&, 113 ' say, such that g(vo ;1l.3)

by the definition of a graph.

Now {w I(v , w) e M(m» is an interval ifo

wl ,w2 e (wl(vO,w)e M(m» -+ [Wl ,W2] c (wl(vO,w)e M(m)} •

wl ,w2e(w/(vo,w)e M(m» -+ (vO,wl)e G(11l ),(Vo,w2)e G(~),11l,~em ,

by definition of M(m). Thus [G(~),G(~)] c M(m) •

Now {w I(v0' w) e [G( 11l ),G(~)]} = [wl , W2 ] by Theorem l.l.

But (w I(vo' w)e [G( 11l ),G(~)]) c (w I(vo' w)e M(m)}. Hence

[wl ,w2 ] c (wl(vo,w)e M(m)} •

that is, M(m) is a band. Now if TIl.'~em, then G( TIl)'G(~) c M(m),

by definition of M(m). Hence [G(Tll),G(~)] C M(m) by Theorem 1..3.

the elements of Which constitute the set {TI} , will also be denoted

by {TI} , as is customary.

Lemma l.9 Suppose {TI} is a topological space, and g(v; TI) is

continuous in TI for all v in (v) and all TI in {TI}. If ~ is a

connected* subset of (TI) containing TIl and ~, then

(vo,wo)e [G("1l),G(~)] -+ (vo,wo)e G(~), Tl:5 e .&.

Proof: Assume (vo,wo)e [G(Tll),G(~)] and that g(vo ;"1l ) :s g("b;~).

Then w € [g(v ;Tll),g(v ;n-)] by Theorem 1.1. Since ~ is connectedo 0 0 .~

and-g(v ;"1) is a continuous function of TI, by the generalized Inter­o

mediate Value Theorem [Hall and Spencer, 1955], there is some point

iso-g surface,

Proof: Any two points in ~ are in a connected subset,

no points in cOllllOOn, that is, do not intersect.

We will say that t'M:> iso-g surfaces are parallel if they have

Through any point in (TJ) , not on an iso-g surface C,

sUbset, ~ say, of (TJ) , containing TJl and ~ (any two points in m)

have all iso-g surfaces through points in it intersecting m.

Theorem 1.10 If (TJ) is a topological space, and g{v;TJ) is

continuous in TJ for all v in (v) and all TJ in (TJ), then a suffi-

Corollary 1.11. Any connected sUbset, ~ , of (TJ} ,is band

[G{TJ1 ), G{~)] C M(m), Where TJl'~€ m. Let (vo'Wo)€[G{TJ1),G(~)];

TJl , ~€ m. By the previous lemma, Tl; € ~ such that

(v ,w )€ G{TJ7:). Hence T).,€ C(v ,w). By hypothesis, C{v ,w )o 0 ~ ? 0 0 0 0

intersects m somewhere; say at 1J4. Applying the definition of an

cient condition for m to be band-producing is that some connected

Proof: By Theorem 1.8, m is band producing, if

(Vo'Wo)€ G{ TJ4)' 1J4€ m. But G( TJ4) c M(m), by definition of M(m).

Therefore (Vo'wo)€ M(m).

producing, under the conditions of Theorem 1.10.

namely ~l' of ~ •

Theorem 1. J2

there passes at least one iso-g surface parallel to C.

Proof: Consider the iso-g surface C(Vl'wl ) w'ith equation

g{Vl,TJ1) = wl • Let ~ be a point in (TJ) not on C(vl,wl ).

Now g(Vl;TJ) = g(vl;~) is the equation of an iso-g surface through

I

I.IIIIIIII_IIIIIII••I

~ as ~ satisfies the equation of this surface. It is clear that

g(vl'~) 1= wl ' as if g(vl'~) = wl ' then ~ would be on C(vl'Wl )'

but we have assumed it is not. Finally C(vl,wl ) and C(vljg(vl'~»

must be non-intersecting as it is impossible that ~ should exist

such that g(vl ;Tl.3) = wl and g(vl ;Tl.3) = g(vl;~)' where

g(vl'.~) 1= wl ' as g(v;71) , for fixed TI , is a single-valued function

of v.

Now we give definitions of sides of iso-g surfaces, analogous

to the definitions of sides of graphs in (G).

~(V,W) is the upper side of e(v,w) iff

~L(V,w) is the lower side of e(v,w) iff

~eorem 1.13 Assume (T}) is a topological space, and g(v;T}) is

continuous in 71 for all v in (v) and all TI in (TIl. Also assume

all iso-g surfaces are connected subsets of (TI). Let (C 1 be a set

of iso-g surfaces. Let CR 1 be a set of sides of iso-g surfaces

selected in the following manner. If e(v,w)e (el, then ~(v,w)

or ~L(v,w)efm), but not both, and if !Ru(v,w) or ntr,(v,w)eORJ , then

e(v,w)e (el. Let I = nfm~e CR)}. Then I is g-wise exhaustive.

Proof: If I is the null set, we note the null set is g-wise

exhaustive.

Assume I is non-null. Let TIl be a point in (TI1not in I.

Then by the definition of I, and application of DeMJrgan I s Theorem

I_.IIIIIII

_IIIIIIII-.I

I

I.IIIIIIII_IIIIIII

••I

15

on the complement of an intersection of sets, g(vl,T}l» wI' where

lRL(vl,wl )€ (!R) , or g(Vl;T}l) < wl ' where !Ru(Vl''''l)€ f'm). Assume

g(vl,T}l) > wl ' where lRL(Vl,wl )€ (!R). The argument would be

essentially the same if the alternative held.

Now through ~ there passes C(V2,w2 ), parallel to C{Vl,wl ) by

Theorem 1.12. That is, there is a point ~€ C(v2

,w2 ) such that

g(Vl,'lll) > wl • If C(V2'W2 ) intersects I at any POint, 1'1:2 say,

this means g{v~~) ::: wl from the definition of I. Since C(v2,w2 )

is a connected subset of{T}) , and since g(vl;T}) ~s a continuous

function of TI , by the Intermediate Value Theorem [Taylor, 1955],

there must exist T}3 on C{V2;W2 ) such that g(Vr~) = wl • That is,

C(v2, W2) intersectsc(v1, wl ' at 'I'l:3' contrary to the fact that

C(v2,w2 ) is parallel to C(vl,.wl ). Hence C(V2

,w2

) does not intersect

I anywhere. Thus I is g-wise exhaustive by Theor~m 1.7.

Note that if I in the above theorem is connected, M{I) will be

a band by Corollary 1.11. Thus if a (l-a) confidence region for

T} is of the form described in Theorem 1.13 and also connected, theo

matting based on it will be a (l-a) confidence band for G(T}o)'

There is a connection between the concept of comprehensiveness

defined early in the section and the later theory. To aid in

elucidating this connection we make the following definition.

Given two points, 'Ill' ~€ ('Il), a set of points in (T}) will be

called a link between 'I'll and ~, denoted by L( TIl' ~)(possiblY with a

SUbscript on the L) iff M{L(~,~» = [G(Tll),G(~)].

betvreen every two points in {T}}.

Proof: First we prove necessity. Assume m is band produc-

~1eorem 1.15 If (G(11)I11£{11}) is comprehensive, a necessary and

through the point intersect m.

I_.IIIIIII

_IIIIIIII-.I

(G( 11) 111dT})} is comprehensive, iff a link exists

Proof: If L(~,~) exists, given 111'~ in {T}} , then

M(L(111'~» =U(G(11) 111£ L(T}l'~» = [G(T}l),G(~)].

Hence {G(11)I11E{11}} is, by definition, comprehensive.

If (G(T})I11£{11)} is comprehensive, then given 111,t~€{11) ,

[G(111),G(~)] = U(G(T}) IT}dT})o C (11)) = M( (11}0).

Hence {T}}o is a link between T}1 a.nd ~.

ing. Since {G(11)I11£(11}} is comprehensive, a link, L(~,,~) ,.L ,_

sufficient condition for m, a subset of{11} , to be band produdnv.

is that for any two points 111 , ~£ m, there is some link between

them such that given any point on the link, all iso-g surfaces

Theorem 1.14

exists between every two points, T}l'~' in (T}). If Tl:se:L(T}l'~)'

T}l'~ £ m, then G(~)£ [G('Ill),G(~)], by the definition of

L(T}l'~). And since m is band-producing, [G(T}l),G(~)] c M(~)

by Theorem 1.8. Now if ~£ C(v,w), then (v,w)€ G(T}) by the

definition of an iso-g surface. But G(~) c M(m). Ti,at is,

(V,vT)£ G(T}4),T}4£ m, by definition of M(m). Thus C(v,l·r) intp.rs('(·I·.~

~ at T}4 by definition of C(v, w).

Next sufficiency is proved. We want to show that if

T}l'~£ m, [G{T}l),G(~)] c M{m), as then by Theorem 1.8, !R is band

producing. If (v, w) £ [G( 'Ill),G(~)], then by definition of 1,( TJ1 ,. TJ),

I

I­IIIIIIII_IIIIIIII-

I

17

~€ L(T)l'~) such that (v,w) €G(~). Hence ~€ C(v,w) by

definition of an iso-g surface. By hypothesis, C(v,w) intersects

!R at T)4' say. Thus (v, w) € G( T)4) c: M(!R) •

Corollary 1.16. If some link, L( T)l' ~), between T)1 and ~

in !R , a subset of {T)}, has all iso-g surfaces through it inter­

secting !R , then any link between T)1 and ~ has all iso-g surfaces

through it intersecting !R.

Proof: By Theorem 1.15, !R is band producing.

What we are to prove is that the condition stated in Theorem 1.15

applies to any link between T)1 and ~. Given that !R is band

producing, this was shown in the first part of the proof of

Theorem 1.15.

Let (T) f be the Cartesian product of {T)} with itself. We

will say (T)l' ~) in (T):f straddles C (v, w) iff ~ € lRL(v, w) and

~€ !Ru(v, w). Note (T)1'Tl.I.) straddles C(v, w) iff T)1€ C(v, w).

Theorem 1.17 If L(T)l'~) exists, then

(T)1'~) straddles c(vo,wo) < '> L(T)l'~) intersects C(vo,wo) •

~: (T)1'~) straddles C(v ,w )1 by ~e~nition of straddling,

Wo€[g(vo ;T)l)' g(vo;~)]

! by Theorem 1.1,

(v0'wo) € [G( T)l),G(~)]

! by definition of L( T)l' ~),

there exists ~€ L(T)l'~) such that (vo'wo) G(~)1 by definition 01' C(vo,wo),

~€ L(11:L'~) and ~€ C(vo'wo).

18

Let m2 be the Cartesian product of m with itself.

Theorem 1.18 Assume (G( T}) IT}E(T}}) is comprehensive.

m 1 a subset of {T}} 1 is g-wise exhaustive and band producing iff,

given T}lE(T}}, to s8\Y every iso-g surface through Tl:t is straddled

by an element of m2 is to imply Tl:tEm.

Proof: Assume m in (T}) is g-wise exhaustive and band produc­

ing. Let T}l be a point in (T}) such that all the iso-g surfaces

through it are straddled by elements of m2 • We will show every

iso-g surface through T}l intersects m. Hence since m is g-wise

eXhaustive1 T}lE m 1 by Theorem 1. 7.

Let C(vo'w0) be any iso-g surface through Tl:t. Let (~1 Tl.3 )

straddle C(VO, WO), ~,~E m. By Theorem 1.17, C(VOI W

O) intersects

L(~, ~). Since m is band producing, [G( ~), G(~ )]c: M(m) 1 by

Theorem 1.8. Also M(L(~,~» = [G(~),G(~)], by definition of

L(~, T}3). Thus

m being g-wise exhaustive ... T}4E m1 by the definition of g-wise

eXhaustiveness. That is, L(~,~) c: m. Thus C(vo,wo), by inter­

secting L(~,~), intersects m. This concludes the proof of the

first part of the theorem.

Now assuming the condition stated in the theorem, we will pro1T'r;

m j 5 band-producing by showing that T}l' ~E m...L(~,~) E"!R. If

this is true, then [G(T}l),G(~)]c: M(m), which means m is band

producing by Theorem 1.8. To see that L(T}l,~)Em, first

observe L(T}l'~) exists since we assume (G(T}) IT}E{T}» is

comprehensive. Next we note that Theorem 1.17 tells us that every

I_.IIIIIII

_IIIIIIII-.I

I

I.IIIIIIII_IIIIIIII·I

iso-g surface through any point in L(~l'~) is straddled by .

( ~1' ~) in !R2• Hence by the condition of the present theorem,

L(~l'~)E!R •

Finally we show the condition of the theorem implies !R is

g-w1se exhaustive. Consider ~l such that G( ~l) c:!R. We want to

show ~lE!R. By Lemma. 1.6, any iso-g surface, C(v ,w ) say,o 0

through 'Ill intersects !R at ~,say. Thus C(vo,wo) is straddled

by (~l'~)E !R2, and by hypothesis ~lE!R •

Theorem 1.19 If a link, L(~,~) exists between 'Ill and ~ in

(~) , and G(~l) and G(~) intersect at (vo,wo)' then

L( 'Ill' ~) c: C(v0' w0) •

Proof:

Recall {w \(v0' w) E [G( ~l'G(~)]}

= [min(g(vo,~),g(vo;~»' max(g(vo;'Ill)g(vo;~)]

by Theorem 1.1. But g(vo ;'Ill ) = g(vo;~) by assumption.

Hence (w\(Vo,W)E [G(~),G(~)]} = wo • Now let ~3E L(~,~).

Then G(~) c: [G('Ill),G(~)]. Thus g(vo;~) = Wo ' and by definition

of C(v ,w ), 'Il3

EC (v ,w ).o 0 0 0

In the following development, the terms "infimum" and

"supremum" are used. Their definitions are given in the appendix.

We remark here that if a set of real numbers has an upper bound,

it has a supremum which is also a real number. Similarly if a set

of real numbers has a lower bound, it has an infimum which is a

real number. For the sake of convenience, we will say that if a

19

Hence the conclusions of the theorem follow.

20

of real numbers will have both a supremum and an infimum.

and if it has no lower bound, its infimum is - 00 • Thus every set

I_.IIIIIII

_IIIIIIII-.I

= W, T}E m} = sup g(v ; T}) •m 0

= W,T}E m) = inf g(vo;T}).m ,

sup (w Ig(v0; T})

inf (w Ig(v0; T})

{(v , w) I (v , w) € M(m)} = {(v,w) I (vo' w) E U (G( T}) IT}E m) }00. 0

by definition of M(m).

But {(v ,w)l<v ,w)€u(G(T})IT}€m» = {(v ,w)1 g(v ;T}) = w,T}€m)o 0 0 0

by definition of G(T}) and the set theoretic term "union".

Thus {(v ,w) I (v ,W)E M(m» = (wlg(v ,T}) = W,T}E m}o 0 o.

and (wl(v ,w)€ M(m)} = (wlg(v ,T}) = W,T}E m} •o 0

Now note the following equivalences of notation.

Theorem 1.20 Let m be a subset of {T}} and let v0 be a particular

value of v. Then sup (w I<v , w) E M(m)} = sup g(v ; T}) ando m 0

inf{wl<v ,W)E M(m)} = inf g(v ;T}).o m 0

Proof:

set of real numbers has no upper bound, then its supremum is + 00 ,

Corollary 1.21 If m is band-producing, {w I(v0'w) E M(m) }

is an interval with end points, inf g(v ; T}) and sup g(v ; T}).mOm 0

Proof: Since m is band-producing, {w I(v , w) E M(m» iso

an interval and has infimum equal to inf g(v ; T}) and supremummO.equal to sup g(v ; T}) according to the above theorem.m 0

That inf g(v ;T}) and sup g(v ;T}) are the end points ofmOm 0

21

(w I(v , w) E M(fIt)} f'ollows f'rom the def'initions (to be found in theo

appendix) of' the terms supremum, infimum, and end points.

° along the straight line segment specif'ied. Thenq

g(vo;~l) :s g(vo;~) :s g(vo;~)' ai'.§; E <B(fIt).

Similarly if' g(v ; a) were mnotonic decreasing in a over the0- q

straight line segment ~l'~' there would be a point on <B(fIt),

Let us suppose ~ = ~ = [°1, °2, ... , 0p], a 1 x p vector of'

real valued parameters. Denote a particular value of' ~ by

attaching a subscript to ~ and its individual elements. Thus if' ~

is a particular value of' ~ , then ~l = [Oll' °21, ••• , 0pl].

Since the components of ~ are real numbers, we will consider

the elements of' (~) to be located in p-dimensionaJ. Euclidean

space, EP • We denote the straight line segment in EP between

the points ~l and ~ by ~l'~. ~l'~ includes ~l and ~.

*Theorem 1.22 Let ~ be a particular point in fit, a closed

subset of' (~). Suppose the straight line through ~ parallel

*to the ° axis intersects <B(fIt), the boundary of' fit at pointsq

~l and ~, such that 0ql:S 0q2 :s 0v. Also let g(vo;~) be

a monotonic function of' ° over the straight line segmentq

~1,!3. If' the above is true f'or every point, ~, in fit, then

inf' g(v ,0) = inf' g(v ;~), and sup g(vo;~) = sU!l g(v.Q;~) •fit 0 - <B(fIt) 0 fit <B(m)·

Note the axis merrliioned can vary f'rom one particular ~-point to

is a monotonic increasing function ofSay g(v ; 0)0-

Proof':

another.

I

I.IIIIIIII_IIIIIIII·I

22

namely ~, where g(vo;~) would be of' lesser or equal magnitude

than at ~, and a point on <B(m), namely ~l' Wher,e g(vo;~)

would be of' greater or equ.al. magnitude than at .!?e.Now <B (m) c: m since m is a closed set. Thus f'or every

particular point, ~ in !R, if' the conditions stipulated hold, there

exist points on <B(m) c: m where g(v ; e) is at least as extreme aso-

at.!?e. Thus we need consider only <B(m) in f'inding in£' g(v ; 9). !R 0-

and sup g(v ; 9). That is, sup g(v ; 9) = s'W g(v ; 9), and!R 0 - . m 0 - <B(m) 0 -

in£' g(v ;9) = in! g(v ;9).!R 0 - <B(m) 0 -

I

••IIIIIII

_IIIIIIII

••I

I

I.IIIIIIII_IIIIIIII·I

2. CONSTRUCTION OF CONFIDENCE BANDS FOR

UNIVARIATE cmmATIVE DISTRIBUTION FUNCTIONS

2.1 Theory Useful for Putting Confidence Bands on the Cumulative

Distribution Functions of Transformed ~andam. Variables.

In this section we apply some of the previously developed theory

to the problem of putting confidence bands on the c.d.f. 's of single

random variables. We will show that certain commonly used families

of c.d.f. 's are comprehensive and that certain confidence regions

for the parameters indexing these families are exhaustive and band

producing with respect to the families involved.

Before considering specific distributions we will develop some

results for univariate distributions in general. To this end we

postulate the following situation.

Let Z be a r.v. with known continuous c.d.f., K(z).

Let h(z;~) = x, defined for all real z, be the equation of a

typical member of a set, (h(Z;~)}, of strictly increasing continuous

transformations. (h(z;~)} is indexed by the parameter set {~}.

(It will be evident that {~} is also the index set for each of the

other sets or families defined innnediately below.)

Let h-l(x;~) = Z be the equation of a typical member of the

set, (h-l(x;~)}, of inverse transformations.

Let {J} be the family of graphs corresponding to (h(z;~)} and

thus to (h-l(x;~)} as well. That is,

J(~) = ((x, z) Ih(Z;~) = x} = (x, z) Ih-l(x;~) = z} •

24

We will use the sYmbol X to denote any given one

of the random variablescorreaponding to the transformations

in the set (h(z;~)}.

Let F(x;~) =y be the equation of a typical member of the set,

~(x;~)} , of c.d.f. fa of these random variables.

Let {D} be the family of graphs corresponding to CF(x;~)}.

Now F(xo;~) =Pr{X ~ xo ) =Pr(h{z;~) S xo) =Pr(z S h-l(xo'~»'

since h(z;~) is strictly increasing. That is,

F{x ;e) = K(h-l(X ;e»0- 0-

(If h(z;~) were strictly decreasing we would obtain

F(x ;e) = l-K(h-l{X ;e», and Theorems 2.1 to 2.4, would still hold,0- 0-

with slight alterations in their proofs).

Theorem 2.1 If L{~l'i!e) is a link between ~l and ~ w.r.t.

(h-l(x;~)} , L(~l'2e) is a link between ~l and ~ w.r. t. CF(x;~)}.

Proof: We are to assume M(L(~l'~» = [J<'~l}J{~)] in the

(x,z) plane. Let M'{L(~l)' (i!e» be the matting in the (x,y) plane

based on L(~l'~). We will show that M'(L(~l'~» = [D(~l),D(2e)]'

Which, by definition, means that L{~l'2e) is a link between ~l

and ~ w.r.t. tF(x;~)}.

Let (xo'Yo) be in D(~) c M'(L{~l'~»' Where ~€ L(~l,2e).

Now y = K(h-l(X ;8..» = K(z ), say. (x,z)€ J(a.) c [J(~,),:r(J3n)],o 0 -::J 0 0 o. . -? -... ~

since L(~l'~) is a link between ~l and ~ w.r.t. (h-l(x;~)},

according to our hypothesis. Suppose h-l(X ;e,) < h-l(X ;(2

) •0-..1. - 0 -

(The argument would be essentially the same if the reverse were true.)

Then z € [h-l(x ; e, ), h-lex ;~)] by Theorem 1.1.o 0 -..I. 0 -c;

I

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_IIIIIIII-.I

25

by the previous theorem, a link exists between every two points in

Theorem 2.2 If {J} is comprehensive, (D) is comprehensive.

exists between every two points in {~} , by Theorem 1.14. Therefore,

(J) is comprehensive, a link w.r.t. (h-1(x;~)}Proof: If

h-1 (X ;9,) < z -+ F(x ;9,) = K(h-\X ;e,» < K(z ) = y ,o -.L. - 0 0 -.L. 0 -.L. - 0 0

since K(z) is non-decreasing,being a c.d.f., and similarly

z < h-1 (x ;9n )-+y = K(z ) < K(h-1 (x ;91")) =F(x ;~n).o 0 -r;. 0 0 - 0 -r;. 0 --.c:;

Hence y e [F(x ;9,), F(x ;9,)] and (x,y)e [D(e,),D(~n)], byo 0 -.L. 0 -.L. 0 0 -.L.--.c:;

Theorem 1.1.

Next we show if (xo,yo)e [D(~l),D(~)], then

(x , y ) e M' (L( 9, ,5!n». Now by Theorem 1.1,o 0 -.L. --.c:;

Y e [F(x ;9, ),F(x ;9-,»] = [K(h-1 (x ;9,H,K(h-1(x ;~»].o 0 -..L. 0 --.c:; 0 -..L. 0 -~

Now K(z) is a continuous function of z. Hence, by the Intermediate

Value Theorem, there exists z such that z e [h-l(x ; 9, ), h-l(x ; el") ]o 0 o~ o~

and K(zo) = Yo. Since L(~l'~) is a link beti<teen ~1 and ~2

'\ol.r.t. (h-1(x;~)), there exists ~3e L(~l'l!e) such that

h-1 (X ,e.) = z. But K(z ) = K(h-1 (X ;9~» = F(x ;EL) = y • Thato ...., 0 0 0 -./ 0 ";'5 0

is, there exists ~e L(~l'l!e) such that F(xo;~) = Yo. Thus

(xo,yo)e M'(L(~l'~» by the definition of a matting.

(~) w.r.t. (F(x;.@»). Hence by Theorem 1.14, (D) is comprehensive.

Theorem 2.3 If m, a subset of (~), is band producing w.r. t.

(h-l(x;~»), m is band producing w.r. t. CF(x;~»).

I

I.IIIIIIII_IIIIIIII·I

ing definitions •

26

In the context of this !ection, let C(x,y) be an iso-F surface

and let C' (x, z) be an iSO_h-l surface. That is, we have the follow-

Proof: We want to show (y IF(x ; e) = y, e€ !R} is an interval0- -

for all real x. We are to assume (Zlh-l(x ;e) = z,e €!R} is ano 0- -

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·1I

Hence since z2 e: [Zl'Z3]'

That is, there exists ~ € !R

C(x, y) = (~IF(x;~) = y, !!. € {~}} •

C'(x,z) = (~Ih-l(x;~) = z, !!. d!!.}} •

C(xl,yl ) =U(C'(xl,z) I K(z) = Yl ).

z2dz lz = h-l(xo;~)' ~ € m} also.

such that h-l(xo;~2) = Z2. Thus

K(h-l(xo;~» = K(Z2) = Y2' ~€ m, and

Y2e:{YIK(h-l(xo;~) = y,~ Em} = (yIF(xo;~) = y,!}. €m}.

Theorem 2.4

interval for all real Xo •

Let Yl and Y3' Yl < Y3, be two points in (y IF(xo;~) = Y, ~ €!R} •

Given Y2 € [yl ,y3

], we will prove that Y2e:(yIF(xo;~) = Y, ~ e:!R} •

From this it follows that (y IF(x ; e) = Y, e € !R} is an interval.0- -

Now for given e. Y = F(x ; e) = K(h-l(x ; e»). Thus,- 0 - 0-

(YIF(x ;e) = y,e€!R} = (yIK(h-l(X ;e) = y,e€!R} ,0- - 0--

and we can say Yl = K(Zl)'Y3 = K(Z3) where Zl = h-l(xo'~l)'

z3 = h-l(xo;~3)' and ~l'~€ m. K(z) being a continuous function of

z, we apply the Intermediate Value Theorem in claiming there exists

z2 such that Z2€ [Zl'Z3] and K(Z2) = Y2. zl'z3€(Z Iz = h-l(xo;~),~e: m}

which is an interval by hypothesis.

Proof: Since

27

argument, we can prove the second statement.

Substituting the term "inf" for the term "sup" in the above

x be a fixed valueo

sup F(x ;e) = K(sup h-l(x ;e» ,m 0- m 0-

inf F(x ;e) = K(inf h-l(x ;e»m 0- m 0-

Let m be a subset of (~) and let

r(x ;e) = K(h-l(X ;e» ,0- 0-

sup F(x ; e) = sup K(h-lex ; e)) •0- o-m m

Proof: First we show ~l€ C(Xl'Yl)~ ~l€ U(C' (~,Z) IK(z) = Yl ).

Now ~l€ C(xl'Yl)~ ~l€(~IF(xl'~) =Yl'~ € (~}} by definition of

C(~'Yl)· But F(xl;~l) = K(h-l(xl;~l». Say h-l(xl;~l) = zl.

Then F(~;~l) = K(zl) = Yl , That is,

~l€(~IK(zl) = Yl ' ~ €(~}}CU (C'(xl,z)I K(z) = Yl ) ,

Next we show that ~l€ U(C'(xl,z) IK(z) = Yl)~~l€ c(xl'Yl )'

~l€ U(C'(xl,z) IK(z) = Yl)~ there exists zl such that K(Zl) = Yl '

and ~l€ C'(xl'zl) = (~Ih-l(~,~) = zl}· That is, 11-l(~'~l)= zl.

But F(xl;~l) = K(h-l(~;~l» = K(zl) = Yl •

Thus ~ld~ IF(xl;~) = Yl } = c(xl'Yl )·

Theorem 2.5

of x. Then

K(z) being a continuous non-decreasing function of z,

sup K(h-l(X ;e» = K(sup h-l(x ;e» ,m 0- m 0-

Thus the first statement in the theorem is proved,

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I.IIIIIIII_IIIIIIII·I

28

We note that if z = h(x;~) were a strictly decreasing

function of x, we could prove that

sup F(x ;8) = l-K(inf h-l(x ;8» ,0- 0-

!R !R 1inf F(x ;8) = 1-K(sup h- (x ;8» •

!R 0- !R 0-

At this point, some remarks are appropriate on how one may find

the set of all iso-F(x;8) surfaces, (C(x,y»), say. First find the

set of all iSO-h-1(x;~) surfaces, (C'(x,z)), say, as any iso-F(x;e)

surface is a union of iSO-h-l(x;~) surfaces. To do this, find A',

the set of admissible (x,z), defined as

At = (x,z) Ih-1{X;8) = z, 8 € (~)) •

Then for each (x,z) in A' find C'{x,z).

Next find A, the set of admissible (x,y), defined as

A = (x, y) I F(x;~) = y,8€ (.Q)) •

This may be done by fixing x at Xl' say, and then finding all values

assumed by y = F(~;.Q) = K(z), regarded as a function of .Q, Where

z = h-1(x;~), and.Q can take on any value in (~). By doing this

for all real x, we find A. This atOOunts to transforming A' in the

(x, z) plane into A in the (x,y) plane by the transformation

(x, z) -+ (x, K( z», which is a many-into-one transformation. Having

found A, for every particular point, (~'Yl) say, in A, find all z

such that K(z) = Yl. Finally

C(~'Yl) =U(C' (x1,z) IK(z) = yl ),

and (C(x,y) I(x,y)€ A) = (C(x,y») •

These procedures will be carried out in the sub-sections to

follow.

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••I

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~.2 Linear Transformations

Suppose x = h(z;(a,b» = a + bZ, -co a < co , 0 < b. That is,

x is a linear transformation of z. We will say a is the location

parameter, and b is the scale parameter.

Let us first consider the family (h-l(x;a)} , where b = bl and

a is free to vary over the whole real line. This means

Z = h-l(x;a) = (x-a)/bl • Thus (h-l(x;a)) is a family of parallel

straight lines in the (x,z) plane with x-intercept a, Where a is

the parameter of the family and can be any real number. If we have

two members of this family with equations, Z = (x-al)/bl and

Z =(x-~)/b2,respectivelY, al < a2 ' the region betvTeen their graphs

consists of the graphs of all straight lines in the family with

x-intercept a' such that al :s a I :s a2•

With reference to (h-l(x;a) },

A' = (x, z) liZ = (x-a)/bl , -co < a < co) =

(x,z) I _co< x < co, - 00 < z < oo} •

The equation of an iso-h-l(x;a) surface, C'(~,zl),iS zl = (xl-a)/bl ,

or a = xl-bzl , where (i).'Zl) is any fixed point in the (x,z) plane.

Thus an iSO-h-l(x;a) surface is any point, say al , on the real line.

The upper side of such an iSO-h-l(x;a) surface is the interval

[al,co). The lower side is the interval (- 00 , al J. We note that any

closed interval on the real line can be expressed as an intersection

of these iSD-h-l(x;a) surfaces.

Now let us assume the family (h-l(X;b)} consists of all

functions z = (X-~)/b, 0 < b. That is, a is fixed at al in the

general linear transformation. (h-l(X;b)} is a family of straight

30

lines in the (x,z) plane with x intercept at al and slope lib, where

b is the Parameter of the family. The slope lib may assume any

positive value. Thus, given two members of the family with equations

z • (x-al)/bl and z = (x-al )/b2, respectively, bl < b2, the region

between their graphs is composed of the graphs of a.ll straight lines

in the family with slope l/b', where bl ~ b' =s b2• Therefore the

set of graphs corresponding to (h-l(X;b)) is comprehensive, and by

Theorem 2.2 the set of graphs corresponding to CF(x;b)} is compre-

hensive.

With reference to (h-l(x;b)} ,

A' = (x,z) I z = (x-al)/b, °< b)

= (x,z) I x < ~,z < O} U (al,o) U (x,z) lal < x,O<z} •

Provided Z 4 0, the equation of an iSO-h-l(x;b) surface, C(xl,zl)'

is b = (~-al)/zllwhere (xl'Zl)€A', but zl ~ 0. Thus we see

that C' (Xl'Zl) can be any point, bl , say on the positive b axis.

The lower side of an iSO-h-l(X;b) surface is (O,b J. The upper sideo

of such an iSO-h-l(X;b) surface is [blJ~). Any closed interval on

the positive b~is can be formed as an intersection of such

iSO-h-l(X;b) surfaces. The equation of c'(al,o) is °= (al-al)/b,

which is satisfied by all positive b. That is, C'(al,o) = (bl 0< b).

Both sides of c'(al,o) are the same set, namely c'(al,o) itself.

Finally we consider the family, (h-l(x;(a,b»} of functions

Z = (x-a)/b,-~ < a < ~, 0< b.(h-l(x;(a,b»} is a two parameter

family of straight lines in the (x, z) plane. The x-intercept, a, can

assume any real value. If we fix a at aI' say, we have a one param­

eter sub-familY of straight lines with fixed x-intercept al and

I

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••I

I

I.IIIIIIII_IIIIIIII·

I

31

slopel/b,where b is the parameter of the sub-family. Hence the

slope in this sub-family can take on any positive value. Now any

straight line, in the (x,z) plane with positive slope, falls in one

of these sub-families. Thus we see that the entire family {h-.l(X;(a,b»J

consists of all straight lines z = (x-a)/b in the (x,z) plane, with

positive slope.

If we have any two straight lines in this family meeting at the

point, (xl'zl) say, the region between their graphs can be swept

out by rotating the line with the larger slope in a clockwise direc­

tion about the point (xl,zl) ,until it coincides with the line having

the smaller slope. We note that any position that the line with the

larger slope assumes during its rotation is that of one of the

straight lines in the family {h-lex; (a, b» J being considered.

Similarly if the two straight lines are parallel, the region between

their graphs can be swept out by translating one line until it coin-

cides with the other, and all positions assumed by the translated

line will be those of members of {h-l(x;(a,b»J. Hence by defini­

tion, the set of graphs corresponding to {h-l(x;(a,b»J is

comprehensive and the set of graphs corresponding to ~(x;(a,b»J

is comprehensive.

With reference to {h-lex; (a, b» },

A' = {(x,z) 1-00< x < 00, -00 < z < oo}.

Thus Etl' iSO-h-l(x;(a,b» surface consists of all (a,b) in

{(a,b) I 0 < b} satisfying zl = (xl-a)/b,where (~,zl) is any fixed

point in the (x, z) plane. When we speak of the line constituting

l:tlisO-h-l(x;(a,b» surface, C,(~,zl)' hereafter, we "Till mean that

32

part of the straight line zl = (xl-a)/b,which is in the region

{(a,b) I °< b} of the (a,b) plane. When zl = 0, C,(~,zl) is the

line a = xl' By varying xl' we may obtain any line parallel to the

b-axis. If zl 40, we may write the equation of C' (X]., zl) as

b = ~ _! ,and we see that suitable values of (xl,zl) will givezl zl

any line that is not parallel to either the b or a-axis. However

no value of (~,zl) is such that C'(Xl'Zl) is a line parallel to the

a-axis. Thus the set of iSO-h-l(x;(a,b» surfaces is the same as

the set of all lines that are not parallel to the a-axis.

We can describe the sides of iSO-h-l(x;(a,b» surfaces as

follows. If the equation OfC'(xl,zl) is a = X].' the upper side of

C'(xl,zl) is {(a,b) I a ~ xl' ° < b} , and the lower side is

{(a,b) IX]. ~ a, ° < b}. If the equation of C'(X].,zl) is

zl = (xl-a)/b, ° < zl' then the upper side of C'(X].,zl) is

{(a,b) I b ~ (xl-a)/zl' °< b) , and the lower side is

{(a,b) I (xl-a)/zl ~ b, °< b). If the equation of C'(x1'Zl) is

zl = (xl-a)/b, zl < 0, then the upper side of C'(xl,zl) is

{(a, b) I (~-a)/zl ~ b, ° < b), and the lower side is

{(a,b) Ib ~ (xl-a)/zl' °< b) •

Theorem 2.6 If the family of functions, {h-l(x;(a,b»}, consists

of all functions Z=h-l(x;(a,b» = (x-a)/b, _00 < a < 00, °< b, then

any closed bounded convex region in the half pla.re, b > 0, of the

(a,b) plane, can be formed as an intersection of sides of

iso-h-l(x;(a,b» surfaces.

Proof: In the appendix, it is shown that any closed bounded

convex region in the half plane, b > 0, of the (a,b) plane, can be

I

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_IIIIIIII

·1I

33

side which includes the region. Thus if a bounded region is in the

here. An essentially similar argument would show that

that the theorem will be proved if we can show that the sets

Xl a= - - - inter­

zl zlo < zl' so that

Rotate the line

surface it constitutes. Now all straight lines in the half plane,

b > 0, are iSO-h-l{x;{a,b» surfaces except for lines parallel to

formed as an intersection of sides of straight lines in the half

plane, b > O. If one of these straight lines is an iSO-h-l{x,{a,b)

surface, a side of it is the same as a side of the iSO-h-l{x;{a,b»

the a-axis. The upper side of such a line, b = bl , bl > 0, is

defined to be the set ((a, b) I 0 < bl :s b} and the lower s.:ide is de­

fined to be ({a,b) 10 < b :s bl}. As the region being discussed is

bounded, the entire side of a straight line parallel to the a-axis

{(a,b) \0 < b :s bl , a:s all and ({a,b) I 0 < bl:S b, a:S a} can both

be expressed as the intersection of sides of iSO-h-l{x;(a,b» sur-

lower side, ({a,b) I 0 < b :s bl } , of the line b =bl , there surely

exists al (the region being bounded) such that the region is also in

((a, b) I 0 < b :s bl , a:S al }. A similar statement holds for the

upper side of a line parallel to the a-axis. Hence it is evident

1'lOuld not be required to form the region, but only a subset of that

({a,b) 10 < bl:S b, a:s~} is also the intersection of sides of

iSO-h-l{x,{a,b» surfaces.

faces. The argument proving that ({a, b) 10 < b :s bl , a:S al } is the

intersection of sides of iSO-h-l{x;{a,b» surfaces will be presented

The argument is as follows. Let the line b

sect the line a = al at the point (al,bl ) and letXl a

the slope of the line b = - - - is negative.zl zl

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I.IIIIIIII_IIIIIIII·

I

34

b = ~ - i in a counterclockwise direction about the point (al,bl )x

from its initial position, b = -1 - ~ until it coincides withzl zl

the line a = al • Every position the line assumes in this motion,

but the final position, the line a • al , is an iSO-h-l(x;(a,b»

surface, if we consider only that portion of the position which is

in the half plane, b > O. Denote the intersection of the upper sides

of these iSO-h-1 (x;(a,b» surfaces by ml • Since {(a,b) la S al,O<b)

is the lower side of the iSO-h-l(x;(a,b» surface a=al

,

ml n{(a,b) la ~ al , 0 < b) is an intersection of sides of

iSO-h-l(x;(a,b» surfaces. It is also the set {(a,b) 1O<b~1' as all.

This completes the argument.

2.3 Explanation of a General Procedure for Constructing Bands

We are about to consider how confidence bands on c.d.f. 's can

be formed in some particular instances. We will take up in order,

the normal distribution, the exponential distribution, and the uni-

form distribution. As the derivation of every band, except the one

for the uniform distribution with mid-range and range unknown,

follows the same pattern, only the distinctive ele:rrents in this

pattern will be given in each case. The overall pattern and the

general significance of the ele:rrents in it will be given beforehand,

innnediately below.

1. K(z), a known c.d.f.,will be designated.

2. Next will be presented!l., the unknown parameter vector, con-

sisting of an unknown location parameter, a, or an uru:nown scale

parameter, b, or both. Also shown here will be x = h(z;!l.) = a+bz,

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_IIIIIIII.,I

for

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I.IIIII,II)I·eIIIIIII••I

35

a linear transformation of z, and z = h-l(x;~), the corresponding

inverse transformation.

3. The c.d.f., y = F(x;~) of the transformed random variable, X ,

will be given. The object of the whole procedure, of course,is to

put a confidence band on D(e ), the graph of F(x;e ), the true c.d.f.,-0 -0

where e is the true value of the parameter vector.-0

4. (C(x,y»), the set of iso-F(x;~) surface~will be described.

5. We will assume a random sample of size n is taken from a

population with c.d.f. F(x;~). Based on ~ and ~, estimates from

this sample of ao and bo respectively, a closed connected confidence

region, m, for (a , b ), with confidence coefficient c, will beo 0

exhibited. We note here, that if we let M(m) = U(D(~) I~ Em), then,

since m is connected, M(m) is a band from Corollary 1.11.

6. It will be shown that m is the intersection of sides of

iso-F(x;~) surfaces. From this, in accordance with Theorem 1.13, we

conclude that M(m) is F(x;~)-wise exhaustive. As remarked at the

beginning of Section I, this means that M(m) is a confidence band for

D(e ) with confidence coefficient c.-0

7. The boundary, CB (m) , of m will be given.

8. The point in (e) , at which sup (x~a) occurs vTill be found- CB(m)

every real value of x. Similarly for inf (x-a)CB(m) b

The significance of this is apparent from the following consider-

ations. Since M(m) is a band, (y I(xl'~) EM(m») is an interval with

end points, inf F(~;e) and sup F(x1e), according to Corollary 1.21.m - m - (xl-a)

From Theorem 2.5, we know sup F(xl;e) = K (sup -b-) and- m

Mean and Known Standard Deviation

2.4 Normal Distribution

2.4.1 Band Construction for a Normal Distribution with Unknown

Let 0"0 be the known value of the standard deviation.

I

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elIIIIIII.,I

(~-a) (~-a)'inf F(xl ;2) = K(inf --:s-). But -b-' if considered as eitherm m (xl-a)

a function of a or b, is monotonic. Hence b is a

specific instance of g(v;~) in Theorem 1.22. Applying this Theorem,~-a xl-a ~-a ~-a

we have s~p -b- =m~ -b- and ~f -b- =m~1) •x -a

Thus sup F(~;e) = K (sup -t-) and!It - m(m) x -a

i.n1' F(xl;!l) =K( inf -t- )·m m(m)

(Xl-a)So we see that given the point in (e), at which sup -b- occurs,

-(x -a) m(m)and the point in (e) at which in! 1 occurs, we can construct

- m(m) """i)M(m) , the confidence band on D(~).

9. To summarize, M(m) will be concisely described.

x= IJ.+O"Z.o

Z = (x-I,J.)/O" •o x

3. F(x;e) = 1 I exp(-(t-IJ.)2/2~0' ) dt.- ~O" - 00

o

4. (C'(x,z») is described in Sub-section 2.2.

A = ((x,y) I -00< x < 00, 0 < Y < 1) •

If (x1!Yl)€ A, then C(Xl,yl ) = C' (x1,zl)' where zl is the

unique value of z such that K(zl) = Yl • Now for every C'(x,z)

J. z1. K(z) =.j'!r 1_ 00 exp(-t2 /2)dt, .. 00< z < 00 •

2. 2 = IJ. , - 00 < J..L < 00 •

Then lR=(Il:L' ~], where

•

x = lJ. + (j z.oz = x-1Jocr-

Let J.1 be the known value of the mean.o .

2. e =(j', 0 < (j' •

and Unknown Standard Deviation

1. K(z) = 1 J z exp(-t2/2)dt, - 00 < z < 00.,J2; - 00

x-f.!.inf occurs at ~.

CB(lR) (j'0

37

8. supCB(m)

in {C'(x,z)}, there exists C(x,y) such that C(x,y) = C'(x,z).

Thus {C(x,y)} = {C'(x,z)} • From Sub-section a. 2, {C(x,y)} is the set

of all points, taken separately, on the real lJ.-line.nE x.

... . 1 J.'" J.=5. Let a = x = n •

( ) l+cLet ~l+C) be such that Pr e ~ ei(l+e) =2 where e is

a normal r. v. with zero mean and unit variance.(j'

- 0III = x - ei(l+C) r ,

'\In(j'

- 0~ = x + e~l+c) Jfi ·

6. lR is the intersection of the side (- 00 ,~] of the iso-F(x; lJ.)

surface, lJ. = ~, and of the side [~, 00) of the iso-F(x; lJ.) surface,

2.4.2 Band Construction for a Normal Distribution with Known ~an

II.II,,.IIIII_IIIIIII••I

n-l

I

-­I,I,I

IIII-,IIIIIII-,I

•

,

•

occurs at 0'1 •

occurs at 0'2 '

Pr(~~l ~ X;-l,i-(l+C»'" l~

Pr(~-l ~ X;-l, i(l-c) )= \~c ,

7. m(m) = {O'l' 0'2} •x-~

8. Ifx<~, su)0

- 0 m(m ---a-x-~

inf 0

m(m) (j

4. (C' (x, z)} is described in Section 2.2.

38

Let ~-l, i{l+c) and ~-l, i-(l-c) be such that

separately, on the sitive O'-line, and the positive O'-line itself.

1: (xi

-x)21\ i=l5. Let·o ... s = -----

6. mis the intersection of the side (0,0'2] of the iso-F(x;O')

surface, 0'''' 0'2' and of the side [0'1,00) of the iso-F(x;O') surface,

where X;-l is a -f?- r.v. with (n-l) degrees of freedom.

(n 1)82Then m'" ,[0'1,0'2] where 0'1 = - ,

~-l,-Ml+C)

(n-l)s2

~-l,i(l-C)

A= «x,y) Ix < 1\' 0 < Y < i-} U(I\,i)u «x,y) 11\ < x, ~ < Y < l} •

If (~,Yl) € A, then C(~'Yl) ... C' (JI].' Zl) where zl is the unique

value of z such that K( Zl) = y1. For every C' (x, ,z) in (C' (x, z) },

there exists C(x,y) such that(C(x,y)}= (C'(x,Z)} •

From Sub-section2.2( C(x,y)} is the set consisting of all points, taken

4. (C' (x, y)} is described in Section 2.2 •

A = (x,Y)I-oo<x<oo, O<y<l}.

If (~'Yl)€ A, then C(~'Yl) = C'(~,zl),where zl is the

unique value of z such that K(zl) = Yl • For every C' (x, z) dc' (x, z) }

there exists C(x,y) such that C(x,y) = C'(x,z). Thus

(C(x,y)} = (C' (x, z) lFrom Sub-s ection 2.2, (C(x,y)} is the set of all

lines in the half plane (Jl,O") 10 <O"}, that are not parallel to the

Jl-aXis. That is, (C(x,y)} is the same as the set of iso-11-1 (x, (a, b»

2. e = (1l,0"), - 00 < Il < 00 , 0 < 0" •

39

occurs at 0"1'

X-Ilo-0"

Z

J exp(-t 2 /2)dt , _00< Z < 00 •- 00

supCB(m)

If Il < x,o

K(z) = ...h..~

x=ll+az.

1 x (t-"'2F(x; (Jl,O"» = - J exp( - .::.:::.J::.L) dt •

..[2; 0" - 00 2cP

Mean and Unlmovffi Standard Deviation

1.

2.4.3 Band Construction for a Normal Distribution with Unknown

iso-F(x;(Jl,O"» surfaces.

surfaces dealt with in Theorem 2.6. Hence from Theorem 2.6, we may

say any closed bounded convex region in the half plane, 0 < 0", of

the (1l,0") plane can be formed as an intersection of sides of

II.IIIIIIII_IIIIIII,.I

surfaces as pointed out in the fourth part of this SUb-section.

As such it can be formed as an intersection of sides of iso-F(x;(IJ.,O'»

7. CB(m) is the union of the following four sets, (B1,lB2,<B3, and

m= (IJ.,O') Ix - e -M1+c1

)(j/.[n S IJ. ::s x + e-M1+C1)(7;'J~, 0"1 ::s 0" ::s 0'2)

is a confidence region for (1J.0,0"0) with confidence coefficient cl c2

[M:>od, 1950]. mis shown in Figure 2.1.

I

.­IIIIIII-,IIIIIII

••I

40

«IJ.,O") ,I - e-M1+c1

) = .[n(x-IJ.)/O" , 0"1< O";S 0-2 ) •

«IJ.,O") 1 e-M:r.+c1

) = .In(x-IJ.)/O'' , 0"1:s.0" S 0"2) •

/\ - /\We will say a = x and b = s, here.

Let (IJ. ,0" ) be the true value of (IJ., 0") •o 0

Pr(x - e~l+C ) (jJ[n S 1J.0 S x + e~l+c )O"J..fu)= c1'1 1

Pre 0"1 S 0"0 ;S 0"2) = c2 '

<B =2

(B =1

5.

where 0"1 =~(n-1)s2/~n_1,~ (l:C:> '

0"2 ='~ (n-1)s2/;(2n-1, ~1-c2) •

Since x and s are independent,

6. mis the intersection of the four sets, !Rl'm2,!R3' and !R4,nOlv

to be described. !R1 is the side, (IJ.,O") I - e~1+c1)S J;(x-IJ.)/O",o<O"],of

the iso-F(x;(IJ.,O"» surface, - eJJ1 ) = .[n(x-IJ.)/O" , 0 < 0".2\ +c1

!R2 is the Side'«IJ.,O")~n(x-IJ.)/O"::se-Ml+cl)'0 < 0"), of the

iso-F(x;(IJ.,O"» surface, e-Ml +C1

) =.In(x-IJ.)/O- , 0 < 0". m3

is the

lower side, « IJ., 0") I 0 < 0" S 0"2)' of the line 0" =0"2. m4 is the

upper side, ( IJ.,O-) I 0"1 S 0") of the line 0" =0"1. It is apparent

that !R is a closed bounded convex region in the half plane , 0 < 0" •

I

I.IIIIIII,eIIIIIII••I

.41.

(83 = ((tJ.,o-) 10- = CT2, x - et (1+cI)0-2/~n :s tJ.:S x + ei(l+cI)CT2/~).

(B4 = ((tJ.,o-) ICT = CTI , x - e~l+cI)CT/~n :s tJ. S x+ e~1+cI)0-1/~n).

\\

\

"\

~ ...."._i~ s,L- __',(B4 i

, 1, 1

\ I\ /1- .... ~~---------tJ.

x,O

Figure 2.1 C. do f -wise exhaustive confidence region onparameters (tJ.o' CT

O) of a normal distribution

8. (BI is the closed segment of the straight line

-e-\(1+C) = ~n{x-f.!)/CT between 0- = CTI and 0- = CT2• On this straight2 _ ./~

line segment, :x!-!J. = x-x- e~l+cI)o-l n = x-x _ e J~,CT CT CT ·M1+cI J

wh1ch is a non-decreasing bounded function of CT if x:s x and

a non-increasing bounded function of 0- if x< x. Thus if x:s x ,

occurs at the point (x + e~l+cl

) CT2/~n, CT2 ) = (~, 0-2 ), say,

x~tJ. occurs at the point (X+ei(l+cI)CTI/.fu, CTI ) = (I\' CTI ), sa.y.

Sup X;J.1 occurs at (1J:3,(jl) and inf x;!J. occurs at (I\,a2 ).ffi2 CB2

<B3

and <B4 are closed straight line segments on each of Which

(x-IJ.)/(j is a monotonic bounded function of IJ.. However the end

points of these segments are the points (1J.1' (jl)' ( ~, (j2) , ( 1J:3' (jl)

and (1J.4' (j2) already designated. Thus <B3 and CB4 add no new

points to consider at which (x-IJ.)/ (j may attain a supremum or infimum

(1J.:L,(j1) and inf x;!J. occurs at (~,(j2).

(!l2 1s t~closed segment of the straight line e-Ml+cl

)= .fn(~-H)between (j = (jl and (j = (j2. On this straight lines segment,

x-x + e-Ml+c )(j/J"nx;!J. = 2 (j 1 = x;X + ei(l+Cl)/J"n ,

which is a non-decreasing bounded function of (j if x:s 3C , and a

non-increasing bounded function of (j if x< x. If x ~ x ,

s~~ x;!J. occurs at the point (x-e~1+cl)(j!J"n,(j2)= (1J.4' (j~), say,

and ~~ x;!J. occurs at the point ('X - e~l+cl)(jl/J"n, (jl) = (~,(jl)'

say. If 'X < x, the reverse holds.

I

.­IIIIIII-,IIIIIII

••I

42

sup x-IJ. occurs atCBl (j

sup x-J.1)<B (j

2

x-I\, -a)

2

X-J.1 ():. sup occurs at ~, (j2 •CB(!R) (j

s1ll> x-IJ. = max(sup X-IJ.,lB(m) (j lB

l(j

x-~

= max (­(j2

x-1J.4=-

(j2

Now if x ~ x,

If x < x, the situation is reversed. That is,

over m.

2.4.4 Confidence Regions for the Parameters of the Normal

9. M(m) = (x,y) I(X,7)e: [D(~'0'2),D(~,(J'l)]' x;S x }U ({x,y) I(x,y)e: [D(~,(J'l),D~,(J'2)]'x< x} •

Distribution and Their Suitability for Band Construction

In this sUb-section, we will show that a confidence region

for (Il , (J' ), which we will now derive, is not only a closed boundedo 0

inf x-Il = min(inf !::.!:!, inf x-Il )<B (m) (J' <B (J' ill (J'

1 2x-~ x-~

= min(-, -)(J' (J'

x-~= -.

(J'l

occurs at (~,(J'l) •

:. sup x-J.l: occurs at (~, (J'l) •CB(m) 0' . J

inf x-J.l. = min(inf x-!J:, in! x-H)CB(m) 0' ill 0' <B 0'

1 2x-~ x-~

= mine -, -)0'2 (J'2

x-~

=-.(J'2

:. inf x-f.1 occurs at (""2' (J'2) •CB(m) (J'

:. in! X:-Jl

CB(m) (J'

x-J.l.~~) (J'

If x < x,

I

I.IIIIIIII_IIIIII

I.II

population.

44

these places, is not exhibited explicitly, or proved to have any

I

.1IIIIIII-,IIIIIII

••I

(0 < S)

(S =s 0),,

(The denominator of S is ~n here

rather than ·.In-l. )

= 0

n/2n

convex region in the half plane, cr > 0, and hence suitable for band

If m' is any specified set in the region of positive density

describes it. Since the region, if,in fact, it is referred to in

As a matter of notation in what follows, we will designate a

normal POPulation with mean,lJ. , and variance, 02, as a N(lJ.,o2)

dence regions. The '\ITiter believes that Wilks [1962, p. 383] refers

**to this region (E2 in his notation) and that ~t:>od[1950, p.229] also

construction, but also has minimum area in a wide class of confi-

desirable properties in the present context, this will be done here.

(connecting the variables (lJ., cr) with (X, S) ),

Let X and S be the sample mean and standard deviation

respectively from a sample of size n from a N(O,l) population.

~1e p.d.f. (probability density function) of (X,S) is

f(X,S) = KSn- 2 exp( -n(X22

+ S2»

into Which m' goes under the one-to-one random transformation

X=(x-g) S s

=cr , cr ,or alternatively

sX slJ. = x - S , cr= S ,

of the (X, S) plane, such that Pr«X, S) € m') = fm,f(X, S)dXdS = c >0,

we define m = H(m'), as the region in the (lJ., cr) half plane, cr > 0,

"There K =

It is mathematica1.1.y correct to write

vThere the transformation parameters x and s> 0 are chosen at random

45

x-IJ.Pr«X,S)€ mf

) = Pr« T 'o

n (- )2 2= K s ( -n[ x-IJ. + s ]) , (0 < er)

cf+l exp 202

= 0 , (er:s 0) •

as the observed sample mean and standard deviation respectively from

a random sample of size n from a l( 1J.0,~) popu1.a.tion. In this sub­

section only, we put s =J£ (xi

- x)2/n •i=l

Now mis a confidence region for (1J.0

' ero) with confidence coeffi-

cient c. This may be seen from the following argument.

= Pr«IJ.,er)€ m).o 0

We want to find a region!Ri ' in the (X,S) plane, with prob-

ability c of occurrence, such that H (mi) = mL

has minimum area, over

a.1.l regions m' in the (X, S) plane with probability c of occun-ence.

vThere

Suppose we find a region ml in the (IJ., er) plane with mininn.un

area over all regions, m, in the (IJ., cr) plane such that

I g(lJ., cr)dlJ. dcr = c. Since the transformation we are vlOrking under,m

is one-to-one, a unique region in the (X,S) plane, mi ' exists such

that H(!Ri) =ml • If "Ire can show that

Pr( (X, S) € lRi) = Pr( (1J.0

' ero ) € lRl ), then lRi can serve as lRi previous1¥

I

I.IIIIIIII_IIIIII

I.II

defined. (It is of interest to note that g(fJ.,O") is the "fiducial

density function" of (fJ.,0"). [Kendall and Stuart, 1961]).

We claim that if !Rl is of the form

!Rl = ( (fJ., 0") I k(x, s) S g( fJ., O")} ,

,.,here k(x,s) is a function of (x,s) and J!R g(fJ.,O")dfJ.da" = c, then1

!Rl has minimum area over a.ll!R such that J!ng( fJ., O")d fJ. d 0" = c. We

proceed to prove this.

Suppose !Ro 4 !Rl and let J!R g( fJ., O")d fJ. d 0" = c.o

Denote tne area of a region !R by a(!R) •

Now !Rl = (!Rl - !Ro) U (!Rl n !Ro ) ,

!Ro = (!Ro- !Rl) U (!Rl n !Ro ) •

Hence,

c = J!R g( fJ., u)dfJ. d 0" = J(!R -m )g( fJ., O")dfJ.OO" + J!R rm g( fJ., u)d~O"1 1 0 I 0

= \a(!Rl~o) + \a(!Rln !Ro ) ,

with

with

*en:q>loying the Mean Va.lue Theorem for multiple integrals •

Also

46

I

.1IIIIIII

_IIIIIIII

••I

K= (X,S) I~ .:s

s

I

I.IIIIIIII_IIIIIII••I

47

from the definition of ml

• Hence a (ml~0) .:s a (m0 -Dll ), and since

a(ml ) = a(ml ~o) + a(mln mo ) ,

a(mo ) =a(mo-Dll

) + a(mf mo ) ,

we see that a(ml).:s Ci(mo)' which completes the argument.

Now we show that an ml,of the form indicated, exists and that it

corresponds, under our random transformation, to mi ' a definite

region in the (X,S) plane so that

Pr«X,S)€ mV =pr«~o'O'o)€ ml ) •

We define the region mi in the (X,S) space as

mi = (X,S) IK[c] .:s S3f (X,S») ,

vThere K[c] is chosen so that Pr«X,S)€ mi) = c.

mi = (X,S) IK[c] .:s KSn+lexp (_n(X22+ S22)}

= (X,S) I ~.:s KSn

+l

exp C n(X2/S2 + 1» )s s 2/S2

....K;;;..sn_~ exp Cn(S2X2js2 + S2») •(!)n+l 2(!)2S S

so that Pr«X,S)€ mv = Pr«f.Lo'O'o)€ ml ).

h8

First, we note ml is of the form ((1l,O")/k(x,s) :sg(J.I.,O"»). In

fact, it was to this end that the factor S3 was introduced in the

definition of mi.

Second, Img( Il, O")dJldO" = c, as shown by the following argument.1

Pr«X,S)€ mi) = Im,f(X,S)dXdS = c,1

from the definition of mi.AlSO, Im,f(X, S)dXdS = Img( Il, O")dJlder ,

1 1as we have remarked previously. Hence we have the conclusion,

Img( Il, er)dJldO" = c •1

Thus ml has mininn.un areaamong all regions m in the (Il, er) plane

such that Img( Il, er)dJlder = c. Since Pr«X, S) € mi) =Pr{ (Ilo' ero ) € ml ),

mi = (X,S)/ K[c] :ss3f (X,S»), K[c] chosen so that Pr«x,s)€mi> = c,

is that region in the (X, S) plane of all those with probability c

of occur:rerDe such that the confidence region for (Il ,0" ) it corre-o 0

sponds to, has minimum area. That is, we can say mi = mi' and

ml = mL' where mLand mL are as described previously.

We now show that mL is a closed bounded convex region in the

half plane, 0" > O.

I

.1IIIIIII

_IIIIIIII

••I

K{( Il,er) I =f£l :s g{ Il, er) )

({ er) I K[c] < Ksn

e (-n( (X_Il)2+S2~)Il, s - n+l xP

er 202

{( er) l.tn(~) < .tn( Ksn

) _ n( (i-Il)2+S2) )Il, s - J1+1 202

( - )2 n 2((Il,er) I n X-Il < .tn(..L-)+ .en( ~ ) - ~)

2~ - K[c] cf+l 202

((Il, er) I (X-Il)2 < 202 [.tn( ..!- (!)n+l)_ ns2

])- n K[c] er 202

(( Il,er) I(X-Il)2 :s .t( er) )

=

=

=

=

=

m =L

I

I.IIIIIIII_IIIIIII••I

where J(O") = 2(n+l) a2 In «..!..- )l/n+l !) _ s2. and is definedn K[c] 0"

only for (O"ln ' say, the set of 0" values such that

2. < 2(n+l) -.2 II « K )l/n+l s) ()S o-.fIn -K - = q 0" , say •

- n ~] 0"

We will first Sho"VT that (O"ln is a closed interval, [O"L'O"U] ,

on the positive 0" - axis. This being the case, "t'1e can say that

lRL = (Il, 0") I x~J(O") :s Il ~ x + .Jt(O") , 0 < O"L:S 0" ~ O"U} ,

which is a closed bounded region in the half' plane, 0" > O,is synnnetric

about the line Il = x.We will then show that the part of lRL, lRL+ ' say, on and to

the right of the line Il = x , is convex. It should then be evident

that by a similar argument the part of lRL ' ~_, say, on and to the

left of the line Il = x, is also convex.

If ~+ and !RL_ are both convex, it follows that ~ is convex.

In proving this,we denote a(Il,O") point by 5!. It suffices to show

that if 5!1€ lRL_ and 5!2€ lRL+ ' then the closed line segment connect­

ing them, .@1,2e, is in lRL • To see this, observe that .Ql'i?e

must intersect the line Il = xat a point ~, with 0" - coordinate in

[O"L'O"U]. Thus ~ is in both lRL_ and lRL+, which means

Ql'~ C lRL_C ~ , and ~,2e C lRL+ C lRL • Since .@l'~= .@l~U .@2'~'

we can conclude that .@l'~ C ~ •

We will now study (O"ln. We first observe that lRL contains

an infinite number of points, as it is the image of lRi ' an, infinite

set, under the one-to-one random transformation given previously.

This means that q( 0") 2: s2. for some 0". otherwise !RL would be the

50

point.

null set. We next note that q( 0") is differentiable for all 0" > 0 •

strictly decreases from an arbitarily large value when 0" is suffi­

ciently close to zero. The fact that dq(O") = 0 has only one root,dO"

means, by an application of Rolle's Theorem, that q(O") cannot equal

S2 more than twice.

I

.1IIIIIII

_IIIIIIII

••I

!RL contains more than one

(O")D consists of one point, O"i' such that

/'(O"i) = 0, and ~ contains only one point

But, as remarked before,

Thereforeq(O".) = S2.J.

(IJ.=X, O'=O"i).

only once, we have that

By L'Hospital's Rule, we can then show that lim q(O") = O.0" ~O

n1US q(O") = s2, at least once, by the Intermediate Value Theore~

Differentiating q(O"), we have

dq1~) = 2(~+1)0" [2 /'n( (KK )1/n+1 ~ ) -1 J, (0 < 0") •[c]

~"/~ K ~~~ equals zero once and only once, as J,n«~) ~ )

Assume that q(O") = S2 at only one point, O"i' say. We note

lim q(O") = - 00. Hence q(O") cannot be greater than S2 forO" > 0".,J.

0"-400

for if it were, it vlOuld have to equal s2 again at a second point

with value greater than 0".. So under the assumption that q(O") = s2J.

Thus we can conclude that q(O") = S2 at precisely two points,

0"1' O"U' say, where O"L < O"U. It is now easy to see that q( 0") > S2

for 0" in the open interval (0"1' O"U) and that q(O") < s2 for 0" not in

the closed interval [O"L'O"U]. Thus (O")D:: [0"1' O"UJ.

It remains to show that !RL+ is convex. n1is follows if

,J J,( 0") is a concave function* of 0" for 0" € [O"L' O"uJ • Now,J/,( 0")

as q(cr) =Na2ql(cr) ,

2(n+l) ( ) ( K )l/n+l s )where N = n ' and ql cr = tn K[cJ "iT.

It is helpful to note that qi(cr) = - ~ • Then one can establish

that

To aid in demonstrating this last inequality, let us express q(cr)

51

, we can write

(See Appendix] . d~cr) ()'is concave on [crL, cru J.f < 0 on crL, cru •

for a discussion of this). Since ~t(cr) = Jq(cr)-S2

From this we see that d2.Jt(cr)/da2 < 0 on (crL'OU)' if

[ d2q(cr) _ 1 (dq(cr»)' 2 ] < 0 on (0:,0: ).da2 2{q(0')-s2) dcr L U

dJ~cr) = Na{2ql(cr)-1),

d2q(cr)doe = N(2ql(cr)-3) •

Next, from the fact that q(cr)_s2 < q{cr), for cr E (crL,OU)' we infer

that d2q(cr) _ . 1 (dQ{cr) .\2 < d2q{cr) _ 1 (dq(cr»)2.da2 2{q{cr)-s2) dcr I d:r2 2q{cr) dcr

Substituting for q{cr), dJ~cr), and d:~cr), we have that

I

I.IIIIIIII_IIIIII

I.II

52

zero correlation) for large n. Hence w'hen n, the sample size, is

I

.­IIIIIII

_IIIIIIII

••I

o<a}

< ql(a) [2q~(a)-3ql(a)-2qf(a)+2ql(a)-!J< q~(a) [ql (a)+ ! J< 0 , for ae( aL,aU) •

.in {( \..l., a) Ia)(»), and hence suitable for our purposes.

< 0 on (aL, au), and the proof that !RL is a closedd2.JJ(a}doe

convex region

and c is the confidence coefficient for MA •

Let us assume the ellipse is entirely within the half plane, 0 < cr •

We will derive a formula giving ~~f X;\..l. == \(x), say, and

sup X-\..l. = ~(x), say, for each real x. Then by Theorem 2.5, our~'tA a

confidence interval on F(x;(\h,ao» for any x is

Where s =j~ (X.-X)2/n , a2 =s2-y2 /n, b2 =s2-y2 /2n,1. =1 1. "2, c "2, c

For the normal distribution, the maximum likelihood estimators

of \..l. and a have an asymptotic bivariate normal distribution (with

bounded convex region in the half plane, a > 0, is completed.

Thus

sufficiently large, we may consider using the "asymptoticaJ.ly

smallest confidence region from large samples", [VliThs, 1962], for

(\..l.o' a0). This region, !RA' say, which is also referred to as the

large sample confidence region, is tIle intersection of the half

plane, 0 < a , with an ellipse. As such it is a closed bounded

I

I.IIIIIIII_IIIIII

I.II

53

We will use some geometric reasoning to derive far.mulas for

A.r. (x) and \J(x) for any given x, :llJ. say.x -Il

Consider the family of lines, ~ = A, - 00 < }.. < 00, in the

(1l,0') plane where}.. is the parameter of the family. This is the

family of all lines in the (Il, a) planes with Il-intercept at xl" Nowx -Il

if we can find a line in this family, 'J. =~ , which is tangent

to mA

, then mA

lies entirely on one side of :llJ.~Il = \. This

follows from the theorem that if a line is tangent to a convex region,

then all of the region lies on one side of that line. If mA lies inx -Il

the upper side of the line ~ = \ ' this rreans, from the defini-

*tion of upper side of a line, that

~-IlmA C «Il,er) 1\0' + Il oS xl' 0' > o} = «1l,0') 1\ oS~ ,a> o} •

* ~-IlIf lRA lies in the lower side of the line cr- = \ ' this means that

~-IllRA C «1l,0') 1:llJ. oS \0' + Il, a > o} = (Il,O') 17 oS \' 0' > o} •

x -IlThus, whatever side of ~ = \ ' mA lies in, \ is a bound for the

x -Il -set (}..I~ = A, (Il, a) € m

A). Now since a tangent line to lRA inter-

sects the boundary of mA

, which is in mA ' a tangent line to mA

in­

cludes points in !RA• Hence if '\ is a lower bound forx -Il

(}..I ...l.- = }.., (Il, 0') € mA), then '\ = A.r. (:llJ.) , and if \ is an uppera :llJ.-Il .

bound for (}..I~ = A, (Il,a) E:mA), then,\ = ~(~). That is, if wex -Il

find a value of \ such that ~ = \ is tangent to !RA, we have

found either A.r.(~) or 'u(xl )·

From (2.3), we have

Also (lJ..l'0"1) must be such that

54

I

.1IIIIIII

_IIIIIIII

••I

•

(~_X)2 (0"1-8 )2 .+ - =1.

a2 b.2

(2.1)

We now solve the system of equations (2.1),(2.2), and (2.3) for A

in terms of x, s , a2 , and b2 • From (2.2) we have

(lJ..l' 0"1) is(l.l:J. -x)

(0" - s) b2 b2= (Il-x) ,

(O"looS ) a2 (O"l-s)

we have 1 b2 (I\-X)

'\ = ,a2 (0"1-8 )

(2.3 ) '\= a2 (O"l-S)or

b2 (lJ..l-X)

Let US translate the requirements on '\ into algebraic

equations. If the line ~~lJ. = '\ is tangent to mA at the point

(lJ..l' 0"1)' this means (lJ..l' 0"1) must satisfy the equation of the boundary

of !RA

, which is the equation of the boundary of an ellipse.

That is,

(0"1-8) = '\b2 (lJ..l-X)/ a2 •

Solving the two equations above for (lJ..l-X) and (O"l-S)' we obtain

~-lJ..l-= '\

0"1

Finally -"i:l must be the slope of the tangent line to !RA at (l.l:J.'0"1).

Since the equation of the tangent line to !RA, an ellipse, at

55

•s (l-~ n-lx~ )'c,c

r+ 0'0z, i-There 0'0 is the known value of the standard

deviation.

x =

Z =

1. K(z) = f Z exp(-t)dt if o ~ z ,0

= o if Z < 0 •

2. e = r , o ~ r •

Initial Point and Known Standard Deviation

a quadratic equation in ~, which, when solved, yields

(xl-x) s ±.[n~-i) ~]2 - (s2_b2)[ (Xl

-x)2_a2]

~= ~-~ •

Substituting the two expressioIll above for (~-x) and (O'l-s )in (2.1)

leads to

Substituting for a2 and b2 , we finally obtain

2.5 Exponential Distribution

2.5.1 Band Construction for an Exponential Distribution with UnknovTn

I

I.IIIIIIII_IIIIIIII·I

If (xl'Yl)€ Al , then C(xl,yl ) = U(C'(xl,z) IK{z) = 0)

= U(C'{xl,z)lz ~ 0)

= U(C'(xl,z) Iz ~ xl/(J'o) ,

since CI (x, z) is non-null only for z ::: xl (fo and ~ ~ 0, being in Al •

Thus C{~'Yl) = {111 = xl-(J'oz, z ~ ~/(fo)

= {11o ::: 1 } •

If (xl,yl )€ A2, then C{~'Yl) = U(C'{xl,z) IK{Z) = 0)

=U(C'(xl,z) Iz ~ 0)

= {111 = xl-(J'OZ' Z < 0 }

= {11 xl ~ 1} •

If (xl'Yl)€~' then C{~'Yl) =U{C'(XtZ) IK{Z) =Yl )

= (117 = Xl-(J'oZl) ,

'i'There zl is the unique value of Z such that K{Zl) = Yr Now for

every non-negative value of 7, "1 say, there exists (xl,yl ) in A3

such that C{Xl,yl ) is the point on the 1-1ine with value "1 •

This follows from the fact that for 0 ~ "1,

S = ({x,z) 10 < Z ~ x/(J'o' 0 < x, x-(J'oz = "1 }

is non-null.

4. A' = {(x,z)lz ~ ~ J •o

{C' (x,y)} consists of all points on the positive 1-1ine.

A = AIU~ U~

,.,here A = {(x,y) Ix ~ 0, y = oj ,1

~= ({x,y) 10 < x, Y = o} ,

~= ((x,y) 10 < x, o < Y < K(x/(J' )} •- 0

3. F(x;1) = 1(J'o

= 0

f x exp{ -<t-1) )dt1 (J'o

if 1 ~ x

if x < .,

,

56

I

.­IIIIIII

_IIIIIIII

••I

I

I.IIIIIIII_IIIIIIII·

I

57

If we select (~'Zl) such that (Xl'Zl)€ S and K(Zl) = Yl , we will

find that (~'Yl)€~ and C(~'Yl) = {rlr =:. r l } •

To be precise, from the definition of 8 we can argue that

o < zl~' 0 < K(Zl) = Yl '

Zl-s..xJ<To~K(Zl) = Yl ~ K(~/O'o)'

o < Xl •

Hence (~'Yl)€~.

80 we see that {C(x,y)} consists of all intervals [71,00) Where

o ~ r l , and of all points, taken separately, on the positive r-line.

5. Let ~=X(l) be the first order statistic from our sample. That

is, x(l) is the smallest value in our sample. Then 2n(X(l)-rO

)/O'o

is distributed as -:f "lith 2 degrees of freedom ~stein, 1960],

6. m is the intersection of the side [0,72 ] of the

iso-F(x;r) surface 7 = 72, and of the side [71 ,00) of the iso-F(x;7)

surface r = 71 •

7. m(m) = {rl' 72 } •

X-78. sup occurs at rJ. •m(m) 0'0

inf x-.1 occurs at 72 •m(m) 0'0

Initial Point and Unknown Standard Deviation

I

.'IIIIIII

_IIIIIIII.,I

if x < r •o

if z < o.

r is the known value of tlE initial point.o

o < er •,= 0

= 0

z =

x = r +erz whereox-ro

er •

3. F(x;er)

4. {C'(x,z») is described in Sub-section 2.2.

1. K(z) = f Z exp( -t)dt if 0 :s z,o

sponding to {F(x; r) ) is also comprehensive.

is comprehensive. Hence by Theorem 2.2, the set of graphs corre-

We note that the set of graphs corresponding to {h-1(x; r) )

58

2.5.2 Band Construction for an Exponential Distribution with Known

A =~U~

where ~ = ((x,y) I x :s ro' Y = 0 ) '.

~ = {(x,Y)lro < x, 0 < Y < 1).

If (xl'Yl)€~' C(~'Yl) = C(~,o) =U(C'(~,z) IK(z) = 0)

=U(C'(XJ.,z),lz:s 0)

= {eriO < er) •

If (xl ,yl ).sA2, C(xl,y1 ) = C'(xl'zl)

vThere zl is the unique value of z such that K(zl) = Yr

For any positive value of er, erl say, we may find (~'Yl)in ~ such that

C(XI,yl ) = {erler =erl ). To do this, choose YI to be any positive

value of y.

2.5.3 Band Constructicn for an Exponential Distribution with Un­

known Initial Point and Unknown Standard Deviation.

59

If K(zl) = Y1, and :x:t = er1z1+rO' (Xl'Y1)€ A2, and C(Xl'Y1) ={erler=~:.J.

Thus (C(x,y)} consists of the positive er-1ine and of every

2(n-l)sewhence !Jt = [er1, er21, where er1 = ,~-2,-M1+C)

2(n-1)Seer2 = •

x2n-2,-Ml-C)

x-r8. If x<rO ' sup

0 occurs at er2 ,- m(lR) er

x-rinf 0 occurs at er1- •

CB(m) er

x-rIf r < x, 0 occurs at er1

sup -0 0" ,x-r

in! 0 occurs at er2er •

9. M(m) = [D( 0"1) ,D(0"2)] •

•

if 0 < Z

if z < 0= 0

1. K(z) = f Z exp(-t)dto

point on it, taken separately.A n

5. Let b =s =~ (x.-x(1»/n-1e i=l ~

2.(n-1)s. er e is distributed as "'I? with (2n-2) degrees of free-

dom if 2.:s n [Epstein, 19601,

6. mis the intersection of the side (0, er2

] of the iso-F (x; er)

surface {erler =er2 } and of the side [er1,oo) of the iso-F(x;er) surface

{erler=er1 } •

7. CB(m) = {er1 ,er2 } •

I

I.IIIIIIII_IIIIIIII·I

A=A:LU~U~

where A:L = {(x,y)Ix :s 0, Y = O) , ~ = ((x,y) 10 < x, Y = O}

I

.­IIIIIII

_IIIIIIII

••I

60

if r.:s x ,

if x < '1 •

Z < o} ,

Z = O} ,

O,:Sr ,0< er •

J x exp( -(~)dt'1 er

1= -er

= 0

=Ai U .A.2 U A5(x,z) Ix < 0,

(x,z) 10 :s x,

(x, z) 10 < x,

Z = (x-r)/er •

x=r+az.

e = (r,er)- ,

AI ­1 -

A2=A3=

4. A I

3. F(x; ('1, er»

where

Z ~ O} •~-r

If (xl,zl)€ Ai, ct(~,zl) = ((r,er) IZI = 0:-' 0,:S '1, 0 < er}

= ((r,er) I er = ~ - L...; O,:S '1, 0 < er} •zl zl

That is, (C'(x,z) I(x,z)€ Ai} consists of alllines with non-negative

er-intercepts and positive slopes in the region ('1, er) 10< '1,0 < er}.

~-rIf (~,zl)€ A2, C'(~'Zl) = C'(Xl,o)= (r,er) I 0 = (f , o.:s '1,0 < er)

= ((r,er) 1 l' =~, 0 < C").

Thus (C t (x, z) I(x, z) € A2} consists of all lines perpendicular

to the r-axis in the region (r,C") 10 :s '1, 0 < C"} • If (~,zl)€~ •

C' (x..,zl)=(r,er)/cr=2_L, 0<'1, O<C"} •.L zl zl -

Thus (CI(x,z) I(x,z) €A3} consists of all lines with positive

slope, negative C"-intercept and with negative slope, positive C"-inter­

cept, in the region fer, C") '10 :s 1', 0 < c-).

In summary, . (C r (x, z)} consists of all lines in the. region

{(r,C") /0 ,:S '1, 0 < C"}, except thOse parallel to the r-axis •

61

= ((7, er) I~ ~ 7, 0 < O"} •

If (Xl'Yl)€~' C(XJ.'Yl ) = C'(:x:t,zl)

where zl is the unique positive value of Z such that K(Zl) = Yl •~ - 7

C(~'Yl) = (7, er) I er = - - , 0 < 7, 0 < O"}zl zl

and is thus any line with negative slope in the region

= ((7,0")1 0 ~ 7,0 < O").

If (:x:t,Yl )€ ~, C(:x:t,Yl ) =C(xl,O) =U(C'(:x:t,z) IK(z) =0)

= U(C'(:x:t,z) Iz ~ 0)

= U(7,er) 17~- oz. 0 ~ 7, 0 < O"}z<O

2(n-l)se

•

~-2,~1-c2)CT. =2

,

A.; = (X, y) 10 < x, 0 ~ Y ~ l} •

If (xl'Yl )€ ~, C(:x:t,Yl ) = C(:x:t,O) =U(C'(~z) IK(z) = 0)

= U(C'(:x:t,z)lz ~ 0).

= U ((7,0") 17= :x:t- O'Z ,O~ 7,0< O"}z<o

((7, er) 10 ~ 7 , 0 < er} •

So we see that (C(x,y)} consists of all regions (7,0) bt~ 7, O<O"},

where 0 ~ 71 ' and of all lines with negative slope in the region

((7, (J) 10 ~ 7, 0 < O"} •

5. Let ~ = X(l)' and ~ = se •

If (7,0") is the true parameter value,000" 0"

Pr(max(X(l)-~, Ml+C1~ ,0) ~ 70 ~ max(X(l)-~,i<l-Cl)~ ,0» = cl '

Pr{O"l ~ ero ~ er2 ) = c2 '

2{n-1)Sewhere er1 =

~-2, -Ml+c2 )

I

I.IIIIIIII_IIIIIIII·I

Since x(l) and se are independent [Epstein, 196C],

is a confidence region for (10

'CTo) with confidence coefficient cl c2•

!R is shown in Figure 2.2.

6. We will deal here only with the case whereCT

~, -M l +cl) 2n ~ x(l)

for all CT € [CT1, CT2]. (However other cases will also produce confi­

dence regions suitable for band construction.)

!R is the intersection of the following four sets, lRl'lR2,!R3, and

lR4• !Rl is the side (1,CT) IX(l)-~,~l+Cl) ii ~ 1, 0 < CT}

of the iso-F(x;(Il,CT» surface 1 = X(l)-~,~l-cl) ;n' 0 ~ 1, 0 < CT •

lR2 is the side (Il,CT) 11 ~ X(l)-~,-Ml-cl) ~, 0 ~ r, 0 < CT}CT

of the iso-F(x;(Il,CT» surface 1 = x(l)- ~,~l-cl)2n' 0 ~ 1,0 < CT.

lR3

is the lower side, ((7, CT) I CT ~ CT2, 0 ~ 7, 0 < 0") of the line

0" = CT2

, 0 ~ r. (As indicated in the proof of Theorem 2.6, such a

set may be expressed as an intersection of sides of iso-F(x;(7,O"»

surfaces.)

914 is the upper side, ((7,0") ICTl ~ 0", 0 ~ 71' 0 < 0"), of the line

0" = 0"1' 0 ~ 1. (lR4 may also be expressed as an intersection of

sides of iso-F(X;(7,O"» surfaces as indicated in the proof of

Theorem 2.6)

I

••IIIIIII

_IIIIIIII-.I

I

I.IIIIIIII_IIIIIIII·I

7. CB(m) is the union of the following four sets, CB1

,CB2

,CB3

, and

CBl = ((1,0') 11 = X(l)-~,Ml+Cl) ~, O'l.:s 0' .:s CT2 ) •

CB2 = {(I', 0') 11 = X(l)-~,Ml-Cl) ~, CTl.:s CT .:s CT2 ) •

CTa CT2CB3 = ((I',CT) IX(l)-;, -!<1+C

1) 2n .:s I' .:s X(l)-~,-!<l-Cl) 2ii '

CT =CT2 ) •CT1

.:s I' .:s X(l)-~,Ml-C1

) 2n '

CT = CT1 ) •

\,\\\

\ (73' CT2)--_._------ ----

Figure 2.2 C.d.f-wise exhaustive confidence region on parameters(I' ,CT ) of an exponential distributiono 0

64

8. (Bl is the closed segment of the straight line

between (j = (jl and (j = (j2. On this straight line segment,

x-z =x-x(l) + ~~,j{l+Cl)(j/n =x-x(l) + ~,i<l+Cl)(j (j (j 2n

which is a non-decreasing bounded function of (j if x ~ x(l) , and

a non-increasing bounded function of (j if x(l) < X.

x-7Thus if x ~ x(l)' ~~p (j occurs at

(X(l)-!~,~l+el)<Tt'n,(j2)= (72,(j2)' say,

inf x-7 occurs at(Bl (j

(X(l)-!~,~l+cl)(jl/n,(jl)= (7l ,(jl)' say.

If x(l) =s x, the situation is reversed. That is,

supx-7

occurs at (71, (jl) ,(Bl

(j

inf X-7 occurs at (72,(j2)·(B (j

1

(B2 is the closed segment of the straight line

7 =x(l)- ~,~l-cl)(j/n

between (j= (jl and (j = (j2. On this straight line segment,

x-r = x-x(l) + t~,~l_CJ)(j/n =X-X(l)+ ~,i<l-Cl)(j (j (j 2n

which is a non-decreasing bounded function of (j if x ~ X(l) , and

a non-increasing bounded function of (j if XCI) < X.

say,

I

••IIIIIII

-IIIIIIII

••I

65

•x-Z4--- 0"1

:. inf x-Z occurs at (74,0"1) •m(lR) 0"

If x.:s x(l)' sup (x-7)<B(~) 0"

x-72= -.0"2

:. sup(x~2,> occurs at (72

,0"2) •

inf X-7 = mine inf(x-7), inf(x-Z) )<B(m) 0" m 0" m 0"

1 2x-71 x-74

= min(-, -)0"1 0"1

these segments are the points (Z4'0"1)' (Z2'0"2)' (1'1'0"1)' and (Z3'0"2)

already designated, CB3

and <B4 add no new points to consider at

which x-l my attain a supreIllU1ll or infiIllU1ll over !n.0"

If x(l) < x, the situation is reversed. That is,

sup (~-Z)occurs at (74,0"1) ,<B2 \. 0"

inf x-Z occurs at (Z,'0"2) •<B

20"

<B, and <B4 are straight line segments on each of which x;Z is a

monotonic bounded function of 0". However, as the end-points of

I

I.IIIIIIII_IIIIIIII·

I

2.6 Uniform Distribution

ing to (F(x; (7,0"») is also comprehensive.

66I

••IIIIIII

_IIIIIIII

••I

•t<z

if z < -t ,if -t ~ z ~ t ,if= 1

= z + t

We note that the set of graphs corresponding to (h-l(x; (7,0") }

x-7 ():. sup occurs at 71,0"1 •CB(m) 0"

inf x-7 = min (inf(x-7), inf (x- Z»CB (m) 0" CB 0" (B 0"

1 2x-72 x-73= min(-, -)

0"2 0"2x-7

=-.2 .0"2

:. inf x-r occurs at (73

,0"2) •CB(m) 0"

Mean and Known Range

l.K(z) = 0

9. M(m) = ((x,y) l(x,Y)€[D(72, O"2hD(74,O"1)J,X ~ x(l) }

U ((x,y) l(x,y)€[D(71, O"1>"D(73, 0"2)]' x(.l) < x} •

is comprehensive. Hence by Theorem 2.2 the set of graphs correspond-

2.6.1 Band Construction for a Uniform Distribution with Unknown

real \-l-line.

4. (C'(x,z») is described in Section 2.2.

67

wo

if x < \-l - 2" 'w wo 0

if \-l- 2"~ x ~ \-l + 2" '

x = \-l +w z, where w is the known value of the range.o 0X-Il

Z = w •o

If (xl'Yl)€~' C(xl,y1) = C(xl,l) = U(C'(x1,z) \K(z) = 1)

= U(C'(x,z) Ii ~ z)

= 1

2. .@ = \-l , (- 00 < \-l < 00) •

A =A1U~U~where Ai = «X,y) 1-00 < x < 00 , Y = o} ,

~ = (x,Y)I-oo<x<oo, O<y<l},

~ = «X,y) 1-00 < x < 00 , Y = l} •

If (xl,yl )€ Al , C(Xl,yl ) = C(xl,O) = U(C'(Xl,Z)\K(Z) =0)

=U(C' (Xl' z) Iz ~ -i)= LJ 1 (1l1!J.= x.. -w Z}Z~-2 w J. ,0

= (\-ll~+ i ~ \-l}.

Thus (C(X,y) I(x,y)~) consists of all intervals of the form

[~.ll' 00) where III is any real number.

If (xl,yl )€ ~, C(xl,yl ) = C'(Xl,yl )

where zl is the unique value of z such that K(Zl) = Yl •

Thus C'(Xl'Yl ) = (1l11l = ~-~zl ) ,

and we see that {C(x,y) I(x,y)~) consists of all points on the

I

I.IIIIIIII_IIIIIIII·I

68

= U h.d fl = x.. -w z}t<z 1. 0

w

= {fllfl~~ - fl ·Hence (C(x,y) I(x,y) € A:3) consists of all intervals of the form

(-00 , ~ ], where ~ is any real number.

Thus {C(x,y») is composed of all points, taken separately, on

the real fl-line, together with all intervals on this line of the

form [fll,oo) or (-00, ~], where ~ is any real number.

" x'1'+ XI ... )5. Let a = m =~ , where x(1) and X(n) are the 1st

and nth order statistics from our sample. That is, x(1) is the sma.ll­

est value in the s8JlJIll~and x(n) is the largest value in the sample.m-fl

Then 7 = M, say, is distributed like the sample mid-rangeo

in samples of n from a population with c.d.f., K(z).

p.d.f. of M = n(1_2IMl)n-l if IMI ~ t ,= 0 otherwise , [Carlton, 1946] •

Let Mi(l+C) be such that Pr(M ~ M}(l+c .)= ~l+c) •

Then Pr(-M}(l+c) ~ M)= ~l-c)

since the p. d. f. of M is symmetric about zero.

We put m= [1-I1'~]

where ~ = m - M~l+q,Wo'

U-.. = m + M.!.(.. ..)W •'c 2 .L.+C 0

6. mis the intersection of the side (_oo,~] of the

iso-F(x;fl) surface, fl = ~, and of the side [~,oo) of the

iso-F(x; fl) surface, fl = ~ •

7. CB (m) = {~, '"'2) •

I

••IIIIIII

_IIIIIIII

••I

4. {C'(x,z)} is described in Section 2.2.

2.6.2 Band Construction for a Uniform Distribution with Known---- --,----

~,allx.

1\' all x •

wif x < I..l.o - 2 ''f w w~ I..l. --2<x<1..l. +-2'o - - 0

'f w~ I..l.o + '2 < x •

sup (x-H) occurs atm(!R) Wo

in£' (x.-H) occurs atoHm) lJ)o

9. M(!R) = [D(I\),D(~)] •

Mean and Unknown Range

l. K(z) = 0 if z < - 1 ,-K(z) = z + 1 if -1 < z < 1 ,- -K(z) = l if 1<z •

2. B = w , (0 < w) •

= l

8.

3. F(x;w) = 0X-I..l.o 1

=-+'2W

x = I..l. + W. z, where I..l.o is the known value of the mean •X~1..l. .

oz =-- •w

A = Al~U~UA4~

where ~ = {(x,y) Ix < l..l.o'Y = 0

~ = ({x,y) Ix < I..l.o ' 0 < Y < 1 ) ,

~ = (l..l.o,1) ,

A4 = (x, y) Il..l.o < x,1 < Y < l) ,

A5

= (x,y) Il..l.o < x,Y = l) •

If (Xl'Yl)€~' C(~'Yl) = C(xl,O) = U(ct(~,z)IK(Z) =0)

=U(C'(~,z) Iz ::;: -1)~-I..l.

= U (wlw = 0)z::;:-i Z

I

I.IIIIIIII_IIIIIIII·

I

70

= (wI2(~O-~) ~ w} •

Thus (C(x,y) I(x,y)€~} consists of all intervals of the form

[wl,oo) where ~ is any positive number.

If (~'Yl)€~' C(Xl,Yl ) = C'(Xl'Zl) where zl is the unique

negative value of Z such that K(Zl) = Yl'

x -~

and C(Xl,yl ) = (wlw = lzlO ) •

We see that {C (x, y) I(x, y) € ~} consists of aU points on the posi­

tive w-line.

= c(~o' t) =U{C' (~o' z) IK(z) = t>= C,{~ ,0)

o

= {wlo < w} •

Thus (C(x,y) I(x,y) € A:3} consists of the positive w-line.

If (xl,yl )€ A4, C(xl,yl ) =C1(xl,zl)

where zl is the unique positive value of Z such that K{Zl) = Yl •~-~o

Thus C{~'Yl) = {wlw = zl }

and we see that {c (x, y) I(x, y) € ~) is made up of all points on the

positive w-line.

If (~'Yl)€ A5, then C{Xl,yl ) =C(xl,l) =U{C'{xl,Z) IK(Z) = 1)

= u(c'(xl,z) Ii ~ z)

= U (wlw = (x.. -~ )/Z)l.<z .L 02_

= (wlo < w < 2(xl -1l » •- 0

So {c (x, y) I(x, y) € A5

) consists of all intervals of the form

(0, ~] where ~ is any positive number.

I

••IIIIIII

_IIIIIIII

••I

71

otherwise.

Pr(R S Riel-C»: i(l-c) ,.

Pr(R < RU.l » = i(1+C) •- 2\; +c

= 0

X·,!Josup --,:r (>,:curs at UJ.,m(m)

X-Iloinf' --w occurs at w2 •

m(m)

9. M(m) = [D(Wl ),D(w2)] •

If ~o < x

w =~ •

7. m(m) = {~,~}.x-

8. If x < ~ sup _ ..2. occurs at w2 '- 0'm(!)1) u,

X-Ilinf -~ occurs at tLl •

<B(m; .l

To summarize, (C(x,y)} is made up of all points, taken

separately on the positive w-line, along with the positive w-line

itself', and all intervals on this line of the form [~, 00) or (0, w1

] ,

where ~ is any positive number.

5. Let ~ =r = X(n)-X(l)

Then R = ~ has the same distribution as the sample range in

sa.nu>les of n from a population with c. d. f. K(z).

p.d.f. of R = n(n_l)Rn-2 (1_R) if 0 S R S i ,

Let R be such thatMl-c)

Let R!<l+C) be such that

We say that m = [~,~]

where w - r/R1 - ~l+cL',

w2 = r!R-Ml_c) •

6. mis the intersect. 0\1 of the side (O,w2] of the iso-F(x;w)

surface w = w2

' and 0; tb sinA ::~1.' 00) of the iso-F(x;w) surface

I

I.IIIIIIII_IIIIIIII·I

2.6.3 Band Construction for a Uniform Distributicn with Unknown

Mean and Unknown Range

·l. K(z) = 0 ifz<-t ,=z+~ if--t<z<~ ,- -= l if ~ < z •

2. G=(f!,w) , (_ClO <f! < ClO, o < W) •

x =f! + lI1Z •

3. F(x;(f!,W»= 0 if x < f! - ~,

X-H 1 Of W W= W +"2 J. f!-2':Sx':sJJ.+2"

l Of W= J. f!+2'<x.

4. {C' (x, z») is described in Sub-section 2.2.

I

••IIIIIII

_IIIIIIII

••I

•X-f!

Z =W

A = AlU ~ U~

where ~ = «x,y) 1- co< x < ClO ,Y = 0) ,

~ = (x,y) I-ClO < x < ClO ,0 < Y < l) ,

~ = (x,y) I-ClO < x < ClO ,Y = l) •

If (Xl,yl )€ Al , C(~'Yl) = c(~,o) = U(C'(~,~IK(Z) = 0)

= U(C'(ilz) Iz ,:s --t)~-JJ.

=U((f!,w) 1- = z, 0 < w)z:s- -t W

= ((f!, w) 10 < ~ 2 (f!-~») •

!'Chus (C(x,y) I(x,y)€ Al ) consists of all regions (JJ.,w) 10< w<2(f!-~»)

where ~ may assume any real value.

If (~'Yl)€~' C(~'Yl) = C'(xl,zl)

where zl is the unique value of z such that K( zl) = Yl.

I

I.IIIIIIII_IIIIIIII·I

73

Therefore

We may associate any Yl between zero and unity, and hence any zl

in (-~,~) with any real ~ and have (~'Yl) in ~. Thus

(C(x,y) I(x,y)€ ~) consists of all lines in ((Il,w) 10 < w ), parallel

to the w-axis or with a slope, the absolute value of which is

greater than 2.

If (~'Yl)€~' C(~'Yl) = Ct(~,l) =U(ct(~,z)IK(z) = 1)

=U(ct(~,z)I~~ z)

= U ((Il, w) I(xl -Il}/w = z,o<w).!. < Z2_

= «Il, w) 10<~-2(Il-xl»).

Thus (C (x, z) I(x, z) €~) consists of all regions

«Il, w) 10 < w ~ -2(1l-~»), Where !l:L may take on any real value. In

summary, (C(x,Y») is composed of all regions «Il,w) 10<~(1l-,\») and

«Il,w) 10 < ~ -2(1l-~) l,

where ~ is any real number, and of all straight lines in

«Il,w) 10 < wl which do not have a slope, the absolute value of which

is less than or equal to 2.

5. Let ~ = m, and ~ = r •

m= «Il,w) Ir-2(Il-m) ~ w ~ r/R[c]+2(Il-m), r+2(Il-m)~w~ r/R[cr2(Il-m»)

Where R[c] is chosen so as to give ma confidence coefficient of c.

m is shown in Figure 2.3. (The following sub-section has a discussion

of this confidence region.)

74

6. !)t is the intersection of the following four sets, !)tl' !)t2'

!Jl3, and!)t4 •

!)tl = {(Il, w) Ir-2(Il-m) ~ w, 0 < w} ,

!)t2 = ((Il,w) Ir+2(Il-m) ~ w, 0 < w} ,

!)t3 = ((Il,w) 10 < w~ r/R[c]+ 2(Il-m)},

!)t4 = ((Il,w) 10 < w:s r/R[cr 2(Il-m)}.

We should note that, !)tl is not a side of the iso-F(x; (11, w» sur­

face, ((Il,w) Iw ~ r-2(Il-m)}. The upper side of this surface is the

surface itself' and the lower side is the whole half' plane {(J..L, w) Iw> OJ.

Similar remarks hold for !)t2 •

However we do not require all of!)tl to form!)t , but only a

subset of it that contains!)t. Such a subset can be expressed as an

intersection of sides of iso-F (x; (11, w» surfaces in the following

manner, which is similar to that employed for truncated sides of

lines with slope zero in the exponential and normal cases. Select

a point on the line section {(Il,w) Iw = r-2(Il-m),w > O} such that the

point has a larger w-coordinate than any point in the line section

which is in the botUldary of!)t. Rotate a line about this point in a

counter-clockwise direction from an initial position with absolute

slope value greater than two, to the terminal position w = r-2(ll-m).

All positions in {(Il, w) Iw > O} assumed by the line in this rotation

are iso-F(x; (11, w» surfaces, excepting the terminal position. Each

of these iso-F(x; (11, w» surfaces has a side which contains!)t. The

intersection of all these sides of iso-F(x;(Il,w» surfaces is a sub-

set of !)tl' that contains!)t. Subsets of !)t2' !)t3' and !)t4' that contain

!)t may be formed in a simil.ar manner.

I

••IIIIIII

_IIIIIIII

••I

I

..IIIIIIII_IIIIIII,eI

When we consider !R3

and !R4' it may be more straightforward

to view!R3 as the lower side of the iso-F (x; (I-l, w» surface !R3

'

and!R4 as the upper side of the iso-F(x;(~w» surface !R4.

At any rate, it is clear that !R can be formed as the inter­

section of sides of iso-F(x; (I-l, w» surfaces.

w

w =t-- -a(l-l-m) I Iw =-nr + 2(I-l-m)[c] '\: / [c]

w=r-2(I-l-m) , \ I / ~ = r+2(I-l-~'\ \ I / I

\ \: / I\ , I / II

(0, r ) ','\~ / I~------T-- ---1------

, I I" I I

) \ I I_ _ _ _ _ _ _ __ _ -+ L I _

I ,(J.l:L'~)1 I {~'W-.L)

I I \) / I \-------7--- ---'\------

I' ,/ / 1 ' ,

/1 / I , \/ / I \ ,

/ / I , "I I • \ \-..il1-. "'S"O"'__--4__ !..I__.-.' ,...;"'_.._----

6n,0)

Figure 2.3 C.d.f-wise exhaustive confidence region onparameters (l-lcJwo) of a uniform distribution

75

metrized form. Thus

c.d.f. of a uniform r.v. with mid-range ~ and range tAl in the repara-

T = ~ +~ , and let y = F'(X;(r,T» be the

76

To find M(m), let us reparametrize the uniform distribution

in terms of the initial point, r , and the terminal point T. That

. t tAl~s, pu r = ~ - 2 '

F'(x;(r,T» = 0 if x < r ,

F'(X;(r,T» = X-l if r < x < T ,T-r --

F'(X;(l,T» = 1 if T < x •

!R in the (~, tAl) space is transformed into the region ~, in the (r, T)

space where !R' = {(1, T) III ~ 1 ~ 12 ' Tl ~ T ~ T2 } and

11 = m-r/2R[c] , 12 = m-r/2, Tl = m + ~ , 1'2 = m+r/2R[c] •

If D'(x;(l,T» is the graph ofF'(x;(l,T» in the (x,y)

plane, M(!R') = [D'(11,Tl ),D'[12''t2)]. This is apparent if we note

that D' (r, T) is simply the straight line segment, between the points

(1,0) and (T,l),augmented by that portion of the line y = 0 Where

x < 1 and that portion of the line y = 1 where T < x.

Now fix 1 at any value, r' , say, in the interval [11' r2] and

consider u(D'(r',T) IT€[Tl ,T2 ])= Pl

" say. Pl

, consists of all points

in the closed triangle with vertices (0,13), (Tl,l), (T2,1),augmented

by that portion of the line y = 0 Where x < 13

and that portion of

the line y = 1 'Where T2 < X. We see that

U(Pl

I1€[rl ,12]) =u(D'(r,T) Il€[ll,r2 ],T€[Tl ,T2])

= U(D ' (1, T) I(1, T) € !R')

= M(!R') •

Also u(P1

Il€[1l ,12 ])= [D'(rl ,Tl ),D'(12,T2 )] •

Thus M(!R') = [D'(1l ,Tl ),D'(12,T2)] •

I

JIIIIIII

_IIIIIIII-.I

Under this inverse transformation

above.

[Carlton, 1946].

,

, otherwise

, w = 1'-1' •

= 0

It is clear that M(m) = M(m I ) •

Finally we conclude that M(!R) = [D f (1'1,1'1) ,D I (1'2,1'2)] •

The inverse transformation from (1',1') to (Il, w) is

2.6.4 Confidence Regions for the Parametersof the Uniform Distribu­

tion and T.heir Suitability for Band Construction

u.. = m+ ;'(1- ..!... )'1. Lj. R[c]

(1'2,1'2) goes into (f.2'~) wbere

~ = m + t<~ - 1).[c]

Thus M(m) = [D(l\,~),D(f.2'~)] where ~'f.2,and ~,are as given

In what follows we will find a confidence region for (II w)""0' 0

that is optimal in the same sense that mL was optimal for (IJ. , cr ). , 0 0

in SUb-section 2.4.4. We will use almost the same notation as in

17

Sub-section 2.4.4, the only changes being that (lJ.,w) is substituted

for (lJ.,cr), (M,R) is substituted for (X,S), and (r,m) is substituted

for· (x,s). Adhering to the same procedure as in SUb-section 2.4.4,

we derive !RL in the (IJ., w) space as follows.

Let M and R be the sample mid-range and range, respective~,

from a unifonn distribution on the interval. [-t,tJ. The p.d.f. of

(M, R) is f(M, R) = n(n_1)Rn-2 , 0 .=s R .=s 1-2 1M I .=s 1 ,

I

I.IIIIIIII_IIIIIIII·I

from a random sample of size n from a uniform distribution on the

positive density, but the side formed by R = 0 is not. Any specified

78

Reasoning as in Sub-section 2.4.4, we will find that if

!Rl

= «Il,w) Ik(m,r) .s g{ll,w») where k(m,r) is a function of (m,r),

I

••IIIIIII

_IIIIIIII

••I

!...)wo

, otherwise.

rR=­w

=0

m-l!M = ,w

[ II J.... II + .l...• ] (We assume r > 0.) This~o - ~o ' ~o ~o·

m-Ilfollows from the fact that the p.d.f. of (7 '

o

It is mathematic~ correct to write

is the same as the p.d.f. of (~R).

interval

f f(M,R)dMdR = f g( Il, W)d~W!Ri !Rl

where g(ll,w) = f(m~l!, ;) I *f::~~ I

= n{n-l) rn

-l

, if m _ (~)< Il< m + (w-r)li+l 2-- 2'w

assertion

or alternative~

The region of positive density is all points in the triangle

with sides formed by the lines R = l-2M, R = l-teM and R = 0, where

the sides formed by R = l-2M and R = l-teM are in the region of

r rll=m-i M , w=R' '

where m and r are the sample mid-range and range respectively

region, !R t, with probability, c , of occurrence, in the region of

positive density of the (M,R) plane generates a confidence region,

!R, for (Il , w ) in the (Il, w) plane with confidence coefficient co 0

under the random transformation,

To see this,

1-1 = m+(~r) ,

I

I.IIIIIIII­IIIIIII

••I

79

and f!It g( 1-1, W)d!J.dW = c > 0, then !Itl has minimum area over all1

!It such that f!Itg(I-1,W)d!J.dW = c.

Now let !Iti = ((M,R) \K[c] ~ R3 f(M,R) ),

where K[c] is chosen so that Pr( (M,R) e !Iti) = c.

!Iti = ((M,R) IK[c] ~ n(n_l)Rn+l )

= (M,R) IK[ ] < n(n-l) (!.)n+l )c - W

K= (M,R) \:l£l < n(n-l) (!.)n+l )

~ - ~ W

K= (M,R) l:l£l < n(n_l)rn-l/(!.)n+l) •

~ - R

The image of !Iti in the (1-1, w) plane is

K!Itl = ((1-1, w) I :l£l < g(l-1, w») •

r2 -

We can assert that !Itl = !ItL

' reasoning as we did in Sub-secticn 2.4.4.

However !RL

is not F(x;(I-1,w»-wise exhaustive.

note that !ItL

is a triangle with sides, 1-1 = m- (~r) ,

W~ w(c) , where w(c} is that value of Wsuch that

n-l Kg(l-1, W{c}) = n(n-l) r n+l = :w-.

w(c) ~

(w(crr ) (w{cr r)1-1 being any value in [m - 2 ' m + 2 ] •

Now by Theorem 1.7, !RL is F(x; (1-1, w)-wise exhaustive only if every

point in ((1-1,w») = (I-1,w) 1-00<1-1<00,0 < w) and not in!ItL has at

least one iso-F(x; (1-1, w» surface through it not intersecting !ItL•

In Sub-section 2.6.3, we see that the iso-F(x;(I-1,w» surfaces through( w -r) (w ) -r)

any point (IJ: ,w) in {(1-1,w») where IJ: e(m- {cJ ' m + (c2 ),P P P

80

consist of all straight lines in ( Il, w)} through (IJ, , w ) with a slopep p

greater than 2 in absolute value, and of the regionsw«Il,W) 10 <w < 2( Il-( IJ, - ..E.2 »}

. - p

andw

((1l,W) 10 <W< -2(Il- (IJ, + ..E.2

») .- p

It is geometrica1.1y apparent that if wp > w(c) is sufficiently

close to w( ) , then all iso-F(x, (Il, w» surfaces through (J.I ,w )c ,p p

intersect !RL • Hence, according to Theorem 1. 7, !RL cannot be

F(X;(Il,w»-wise exhaustive.

Suppose we consider regions in the (Il, w) plane of the form

specified by Theorem 1.13. Such regions will be iso-F(x; (Il, w»-wise

exhaustive as asserted in that theorem. Being the intersections of

convex regions, they are convex and hence also connected. Let us

call such regions F(x; (Il, w» ...wise convex regions. (When we speak

of convex regions in this sub-section we mean plane convex regions.)

Now every point on the boundary of a convex region is the

vertex of an angle, the tangent angle* for the region at that

*point, the sides of which are the tangent lines to the convex

region at that point. For general convex regians the slopes of the

tangent lines at a point may be any real numbers and the angle

between the two tangent lines can have any value in the interval

(0, 1!]. For F (x; ( Il, w» -wise convex regions, we Show in the appendix

that the absolute values of the slopes of the tangent lines at a

boundary point must be in the interval [2,00 J.

It is natural to ask,which one,of all F(x;(Il,w»-wise convex

regions in the (Il, w) plane, corresponding to regims of positive

density in the (M,R)

I

.1IIIIIII

_IIIIIIII.,I

I

I.IIIIIIII­IIIIIII,.I

81

plane, with probability c of occurrence, has mininnun area. Call

this region mI. We will show that mI is the region

((J-l,w) !r-2(J-l-m) ::::w:::: Rr

+ 2(Il-m), r+2(Il-m) <w< Rr -2(Il-m»),[c] - - [c]

used as a confidence region for (Il ,w ) in Sub-section 2.6.3.o 0

Actually we will show that ml

has minimum area. over all F(x;{ll,w»-

wise convex regions, m, such that fmg(ll, w)dlJdW=c, g( J-l, w»O for (Il, w) in

m. For purposes of this discussion, let us call such regions c-figures.

If m, has mini:rnum. area aver all c-figures, it certainly has minimum

area over any subset of the set of c-figures to which it belongs, such

as all F(x; (Il, w) )-wise convex regions in the (Il, llJ) plane, Which corre­

spond to regions with probability c of occurrence, in part of the

(M,R) plane where f(M,R) > O.

Since the region in which g(ll,w) is positive does not include

any points such that w< r, the w-coordinates of points in any c-

figure, ~ , say, have a lower bound and hence have a greatest lower

*bound, lA\, ,say. The line w == lA\, is a support line to ~.

Since ~ is closed, the line w = ~ must intersect ~ at same

boundary point, (11)'~)' say. Now the absolute values of the slopes

of the tangent lines to ~ at (I\'lA\,) cannot be less than 2. Thus

from the definition of tangent lines to a convex region at a point,

~ must be entirely within an angle, 3'{11>'~)' say, with vertex at

*(11)' lA\,), the sides of which are rays on the upper side of the line

W = lA\" emanating fram (11)'~) with slopes -2 and 2. Thus the

only point at Which the line w = ~ intersects ~ is (11)'~). We will

refer to (~,lA\,) as the base-point for ~. We see that every c­

figure has a unique base-point.

which is a convergent improper integral, whose value is the sane

I

.­IIIIIII

-IIIIIIII

••I

dw = c ,.. 00 Woo W

( ) n-l f un n-l r w n+l

u w

it is clear that

82

Let us consider :r(,\,~), where (,\,~) can be any point in

the region where g(ll,w) is positive. It is geometrically apparent

that :r(,\,~) is also within the region 'Where g(ll, w) is positive.

Also the length of any line segment on the line w = Ws ' say, in

the angle :r( '\,~) is the difference between the Il-coordinates of

the points where the line w = Ws intersects the sides of the angle.

This difference is

m+ ("'s~"\,) _ (m- ("'.~"\,»)= "'.-"\, ·Hence since g(l.!, w) does nat depend on I.! ,

If Wu is defined by the equation

regardless of the value of ~. This value is unity, when ~ = r

and decreases as ~ increases, tending to zero as ~ tends to

infinity.

and

Since any c-figure with base-point (,\,~) must be in :r(~,~),we

see that no c-figure exists with base-point (,\,~) such that wu<~.

I

I.IIIIIIII_IIIIIII

••I

83

d(~,WU) is the only c-figure with base-point ("'b'wu)' as any

c-figure with base-point (~,wu) must be a sub-set of d("'b'WU)' but

fd ('"1>' Wu)g(ll, w)dlldw = c, so that fs("'b' Wu)g(ll, w)dlldw < c where

S( '"1>' wu ) is a c-figure which is a proper sub-set of 3' ('\, tA\,) •

We will see hereafter that c-figures with a finite area exist.

Since d('"1>' wu ) has an infinite area, it cannot be the region !Itt.

We will nOvT show that if tA\, < Wu ' there is a c-figure, wt.ich

we will call 8(I\.,tA\,), having rnininnm area over all c-figures with

base-point ("'b'~). ~('"1>'~) is the intersection of d(,\,~)

with its reflection in the line w = ~, ("'b' ~), where ~ ("'b' ~), is

chosen so that f~("'b,~)g(Il,W)dIldW= c. To see that a unique

~ ('"1>'~) exists for every base-point ("'b' tA\,), tA\, < Wu ' we note that

if W= Wi 2: ~ and ~i("'b'~) is the intersection of d( '\' tA\,)

with its reflection in the line W=wi ' then

l[ W. (w~) (2w. -~( 2Wi"'~ -w) lf8i('"1>,~)g(Il,W)dIldW = n(n_l)r

n- f~ ur+l dw+f~~ wn+l dW_

J

for fixed ('"1>'~) is a continuous function of Wi' which is zero,

When Wi = ~ and which for sufficiently large Wi attains a value

. () n-lfoo (w-~) .greater than c, smce n n-l r ~ wn+l dw > c, ~f ~ < Wu •

Hence by the Intermediate Value Theorem for continuous functions,

fol\ ( )g( Il, W)dIldW must assume the value of c for some Wi~i '"1>'~ ,

in the interval [~,oo). FurtherIOOre there cannot be two distinct

c-figures ~j('"1>'~) and ~k(I-b''\) such that

It is evident that

84

for in that case, one of the c-figures, say ~j(~'~)' must be a

proper sub-set of the other, ~k(~, ~), as is geometrically appar­

ent in view of their shapes, and this would mean

~(~,~) = Sl(~'~) U S2(~'~)' where Sl(Ilt>'~) = Sl(~'~)

U(Sl(~,~) - Sl(~,~»,

S(~,~) = 81(~,~) U S2(~'~)' where

S2(~'~) = S2(~'~) U(S2(~'~) - $2(~'~» •

I

.­IIIIIII

_IIIIIIII

••I

which contra-

There exists wt such that if we define tre

J~ j (~,~ )g( Jl, w)dJ.,ldw < J~k(~, ~ )g( Jl, W)dJ.,ldW ,

diets our original assertion.

Let us consider all c-figures with base-point (~,~) that are

symmetric about the line Jl = ~. It is clear that ~ (~, ~) is

in this class. We will show that ~ (~, ~) has as small an area

as any other member, S(~,~), say, of the class.

In the appendix we prove the following theorem.

~l(~'~) = ((Jl,w) I<Jl,w)€ ~(~,~),w =s wt ),

~2(~'~) = ((Jl,w) I(Jl,w)€ ~(~,~),Wt < W ) ,

Sl(~'~) = ((Jl,w) I(Jl,w)€ S(~,~),w =s wt },

S2(~'~) = ((Jl,w) I(Jl,w)€ S(~,~),Wt < w } ,

then Sl(~,~) c ~l(~,~), and ~2{~'~) c S2(~'~) •

4 sets,

Theorem 2. 7 •

•

•

,

c = f g(~,w)d~W =&(Ilb, ~)

f g(ll{ w)d~W + f g(ll, w)d~w +81(~,~) &1(~,~)-S1 (~,~)

85

we have c I = f g(ll, w)d~w = f g(ll, W)d~W&1(~,~)-Sl (~,~) S2(~,~)-i92(~'~)

and

and use the Mean Value Theorem :for multiple integrals. We can say

A$ in Sub-section 2.4.4, we denote the area of' a region m by G(m) ,

Now the value o:f the w-coordinate :for any point in

&1(~,~) - Sl(~,~) is less than the value o:f the w-coordinate

f'or any point in S2(~'~) - &2(~'~) •

Thus since g(lJ......,w) < g(1l ,w ) if' w < w , we have A... < ~ ,. p p q q q P '2 - 'l-

and hence a(&l(~,~)-Sl(~'~»~ a(S2(~,~);g2(l-b'~»•

I

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86

Noting that

a{8{~,~» =a{sl.{~'~»+ a{82{~'~»+ a{8l.(~,~)-Sl{~'~»'

a{S{~,~» =a(sl.(~,~»+ a{82{~'~»+a(s2(~',\)-82{~''\) ,

we concl.udethat aw(~,,\».::sa{s{~,,\» , which was to be proved.

We need two IJX)re theorems that are proved in the appendix.

Theorem 2.8. If all horizontal lines intersect a closed convex re-

gion as line segments, which are translated in a direct-

ion parallel. to the horizontal. axis until. their mid-points lie on

a speci~ied vertical. axis, then the ~igure composed o~ the trans-

l.ated l.ine segments is al.so convex.

Theorem 2.2. If the original. convex region in the above theorem

is a c-~igure, then the new ~igure ~ormed fiom it, provided it is in

({Il, w) lui> 0), is also a c-figure with the same area.

I~ we note that the new figure is symmetric about the verti-

cal. axis speci~ied, we may deduce fiom the t-wo theorems above, that

~or any c-~igure, 3' , with base-point (~,,\), there is a c-figure ,

3" , with base-point (~, '\) and the same area as 3', that is

symmetric about the line 11 a ~. Since the area o~ at cannot be

less than the area o~ 8(~,,\}, neither can the area o~ :J.

Thus we have proved that 8{~,,\} has minimum area over aJ..l. c-~igures

with base-point (~,~).

We now see that the c-~igure with minimum area is the region

8 (~, ~) with minimum area. We may f'urther narrow down the group

o~ possible candidates ~or the c-~igure with minimum area i~ we

recognize that D(I-\,w) and D(Il,w) have the same areas i~p p q q

wp = wq • This equality o~ area ~ollows from ~(Ilp'wp ) being

I_.IIIIIII

_IIIIIIII-.I

I

I.IIIIIIII_IIIIIII••I

congruent to $ (J.l. ,W ) if W = W • The congruency follows fromq q p q

the facts that all :r (I-l.t>' ~) are congruent to each other and

$(I-l.t>'~) is defined as the intersection of :r(~,~) with its

reflection in a straight line whose position relative to :r(~,~)

depends on ~, but not on I-l.t> •

Since m is the only Il-coordinate that is associated with

every w-coordinate in the region Where g(.ll, w) is positive, to

form base-point coordinates, we can reduce our task to finding wtsuch that i(m, Wt) has minimum area.

We now show that if ~l< ~2' then a(8(m'~1»<a(8(m, ~2».

To thi.s end, we define the apex-point, (Ils 'ws )' of $(I-l.t>'~) as the

reflec:tion of the base-point (I-l.t>' ~) in the line W= ~ ( I-l.t>' ~).

Let (m,wSl) be the apex-point for $(m'~l) and let (m,wS2) be the

apex-:point for $(m, ~2). From the geometric nature of i(m, ~l)

and $(m'~2) it is clear that wsl < wsl ' if ~l < ~2. other­

wise, $(m, ~2) would be a proper subset of ~(m, ~l) and thus

$(m, Wbl ) and $(m, ~2) could not both be c-figures as they are.

If wsl < ws2' it is not difficult to see that $ (m, ~l)

must intersect ~(m, ~2) in such a way that 'any point in

$'(m, ~~~)...j(m, ~l) has a w-coordinate of value greate~ than the valu.e

of the w-coordinarte of ~y point in $(m, ~l.)-$(m,~2). Thus S(Il, w) is

gIteater at any point in ,em, ~l)-$(m,~2) than at art3 point inI

~(m, ~:L~";$(m, ~2). Since $(m, ~l) and $(m, ~2) are both c-figures

'we can sayJ g(ll, w)d¢w = J g(fl, W)dJ,1dW

$(m, ~1)-Am,~2) $(m, ~2)-$(m,~l) •

under the random transformation

88

Hence by the Mean Value Theorem for multiple integralS,

I

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_IIIIIIII

••I

dRdM

~~ I-2M1 f(M,R)dmdR = n(n-l) 1 1 R

n-2

!Ri+ 0 R[c] (1+2M)

= t-R[~l [(1+R~C] t 1

- tJ ·

r rlJ.=m-RM, w=R •

Thus ~ (m, r) is a confidence region for (IJ. ,w ), and must be theo 0

~(m,r)= ((\.l,w) Ir-2(\.l-m) <W< Rr +2(IJ.-m), r+2(\.l-m)< w<..!:.- -2(IJ.-m)}- - [c] - - R[c]

is the c-figure with minimum area. It is easy to show that ~(m,r)

is the image in the (IJ., w) plane of the region, in the (R, M) plane,

region, !RJ, , we have been seeking.

Let us consider how we could evaluate !R[c]. We note that

both !Ri and f(M,R) are symmetric about the line M = O. If !Ri+

\a(~(m'~1)-~(~'~2» = ~a(~(m'~2)-~(m'~1»

,.,here ~::: \. This implies

is the part of !R.e on and to the right of the line M = 0, we can

put I!R r f(M,R)dmdR = ~ , and solve this equation for R[c].J,+

We find that

So we see that the value of wI, should be the minimum w-value

possible. That is, wI, = r, and

a(~(m,~l)-~(m,~2» ::: U(~(m, ~2)-9(~, ~l» ,

which implies a(~(m'~l»::: a(~(m'~2» •

I

I.IIIIIIII_IIIIII

I.II

Hence fl -A{l-c) ~ n:l

)n-l .l.R c] = (~2 _ ~

[ \.1+R[cl•

3. CRITERIA FOR CONFIDENCE BANDS

3.1 Introduction

The selection of criteria should be made with regard to the

spec1fic purposes for which the band will be used. Then the band

that best satisfies the chosen criterion or criteria should be used.

In this section we will compare some parametric confidence

bands with the corresponding K-S bands with respect to the expected

maximum width of the confidence band and the expected width of the

band at certain points, specified in terms of the parameter values.

Also other criteria will be discussed, which are more appropriate for

comparing parametric confidence bands with each other than with a

K-S band. Such criteria involve the extent of a band in a horizontal

direction rather than a vertical direction, and the total area en­

closed within a band.

Also, using results obtained in this section, we will1\

establish that the distribution of rnaxIF(x;e)-F(x;e) I is parameter-- -0

X

free, in some cases.

3.2 The Maximum Absolute Difference Between Two c.d.f. 's

3.2.1. General Theorl

Let B be a confidence band on G( e). By the definition of-0

a band, (YI(x,y)€ B} is an interVal for every x. Let W(x;B) be

the length of this interval for a given x. We will refer to W(x;B)

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91

as the width of the band B at the point x. AJ..so let us denote the

maximum band width, max W(x;B) by WM • One criterion for a band isx

E(WM), that is, the expected maximum band width.

As an aid to finding E(WM) for the confidence bands con­

structed in Section 2, we will first deal with the maxiImlIll width

of a band, Whose boundary is constituted by 2 c.d.f. 's from the

family of c.d.f. 's to which, we assume, the c.d.f., we are sampling

from, belongs. We suppose that the family of c.d.f. 's, [F(x;.Q)},

is as postulated at the beginning of Section 2. AJ..so x is a linear

transformation of z as defined at the beginning of Section 2.2.

We use the notation of Chapter II Where appropriate.

Now let w(x;a,a',b,b') = F(x;a,b) - F(x;a',b').

We want to find max Iw(x;a,a', b,b') I , the maximum absolute differ­x

erence between the two c.d.f. 'sJ F(x;a,b) and F(x;a' ,b'). It is

evident that this is either I max w(x;a, a', b,b') I orx

lmin w(x;a,a',b,b') I. We observe that under the assumptions madex

in Section 2, w{x;a,a', b,b') is continuous for all x. Also

lim w(x;a,a',b,b') = lim w(x;a,a',b,b') = 0 , as F(x;a,b) andx.... -co x.... 00

F(x;a',b') are c.d.f. 's •

Now some theorems will be stated and proved that will be of

use in deriving results for specific distributions in Sub-section

Theorem 3.1

{x-a)/b ~ (x-a')/b' ~ w{x;a,a',b,b') ~ 0 •

(x-a')/b' ~ (x-a)/b~ 0 ~ w(x;a,a',b,b') •

Proof:

A similar argument holds when (x-a' )/b' < (x-a)/b •

~w(x;a;a',b,b') < o.

(x-a)/b ~ (x-a')/b'~ K(x~a) ~ K(X~~')~F(x;a,b)~F(x;a',b')

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-IIIIIIII

••I

*and a is such that

*= - min w(x;a ,a',b,b').x

( , , ') i *= max w x a, a ,b,b , f a < a ,x -

*= - min w(x;a,a',b,b'), if a < a •x

Theorem 3.2. Suppose b < b' ,

*max w(x;a ,a',b,b')x ~

Then max Iw(x;a,a',b,b')Ix

Proof: We note that

maxlw(x;a,a',b,b')I =max(max w(x;a,a',b,b'),- min w(x;a,a',b,b')~x x x

From this it immediately follows that if a = a*, then

maxlw(x;a,a',b,b') I =max w(x;a,a',b,b').x x

Next we observe that w(x;a,a',b,b') is a non-increasing

function of a. This follows from the fact that F(x;a, b) = K(x~a) ,

where (x-a)/b is a decreasing function of a, and K(z) is a non-

*decreasing function of z. Thus if a < a, we have that

*w(x;a,a', b,b') ~ w(x;a _,a', b, b ') for each value of x.

*Hence max w(x;a,a',b,b') < max w(x;a ,a',b,b') ,x * - x .

- min w(x;a ,a',b,b') < - ~n w(x;a,a',b,b').x - - x

* . * )Since max w(x;a ,a',b,b') = - m1n w(x;a ,a',b,b' ,x ~ x

we have that max Iv(x;a,a', b,b') < - min w(x;a,a', b,b'),x - x

and we concludetbat maxi w(x;a,a',b,b') I = -min w(x;a,a',b,b') •x x

By a similar argument we can prove that

and any a and x. Hence

spcn dence between values in the interval and values the function

93

K{ZL) < Y < K{z ),o 0

(x -a)/b = (x -a')/b' -+F(x ;a,b) =F(x ;a'b').o 0 0 0

We recall that F(x;a,b) = K( (x-a)/b) for any b > 0,Proof:

Theorem 3.3

{xo-a)/b, (xo-a')/b' €(zL'~). If this were not so then either

K«x -a)/b) or K{ (x -a' )/b 1) would not equal y. If a function is000

strictly increasing over an interval, there is a one-to-one corre-

then {xo-a)/b = (xo-a')/b' •

Proof: We are given that F{x ;a,b) = F{x ;a'b') = y •000

Thus K{{x -a)/b) = K«x -a')/b') • y • Now K{z) is monotonic non­000

decreasing on the real line Which implies that

Theorem 3.4 If K(z) is a strictly increasing function on (ZL'~)'

and F(x;a,b) intersects F(x;a',b') at (x ,y ) where K(z-)< Y. < K(~.) ,o 0 LOU

maxlw(x;a,a',b,b') I = max w(x;a,a',b,b'), if a < a*.x x

(x -a)/b = (x -a')/b' -+ K«x -a)/b) = K«x -a')/b')o 0 0 0

-+ F«x -a)/b) = F«x -a')/b') •o 0

takes in the interval. Hence (xo-a)/b,(xo-a')/b' € (~,~), and

K{{x -a)/b) =K«x -a')/b'), taken together imply thato 0

{x -a)/b = (x -a')/b' •o 0

~1eorem 3.5. If K{z) is strictly increasing on (ZL'~)' b < b',

and F{x;a,b) intersects F{x;a',b') at (x ,y ) whereo 0

I

I.IIIIIIII­IIIIIII••I

then x < x ... w(x;a,a',b,b') < 0,o

x < x ... w(x;a,a',b.b') > O.o ' -

Proof: By Theorem 2.3, (x -a)/b = (x -a')/b' •o 0

Thus for x < Xo ' (x-a)/b < (x-a' )/b'... K( (x-a)/b) =:: K( (x-a' )/b')

... F{x;a,b) =:: F(x;a',b') ... w(x;a,a',b,b') SO.

A similar argument will show that w(x ;a,a',b,b') > 0, When x < x •- 0

The following result will be useful When the p. d. f. of z is

symmetric around zero.

Theorem ,.6 If K(-z) = l-K{z), all z, and (~-a) = -(~-a), then

w(~;a,a,b,b') = - w(X1;a,a,b,b').

Proof:

w{x1;a,a,b,b') = F(X1,a,b) - F{X1,a,b')

= K«xl-a)/b) - K{{xl-a)/b') •

w(X2,a,a,b,b') = F(X2;a,b) - F{X2;a,b')

= K{(~-a)/b) - K«~-a)/b')

= K(-{~-a)/b) - K(-(~-a)/b')

= l-K«~-a)/b) - 1 + K«~-a)/b')

= - w{x;a,a,b,b ' ) •

Now we will find What the value of IJWC Iw(x;a,aI,b,b ') I is forx.

any (a,a I,b, b'), if we are dealing with one of the three distribu-

tions considered in the second section. We will consider the normal,

exponential, and uniform distributions, in that order.

We want to provide an exhaustive coverage of all possible

values of (a,al,b,b'). This can be done if we regard the (a,a',b,b')

I

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••I

I

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95

set in the following manner. It is divide~ into two subsets, ~

and ~. U1 contains a1J. values of (a, a' , b, b ') such that b = b'.

~ contains all other values of (a,a',b,b ' ). Then we treat the case

in !1. Where b = b I and a < a I. All points in U1

may be cast in

this form except for those points where the location parameters are

equal and it is appu-ent that maxlw(x;a,a',b,b ' ) I = o. To dispose

of the points in U2 we consider only cases in each of Which b < b ' •

a will change from case to case with respect to its relation to

(al,b,b ' ). It will be am;arent that any point in U2 can be expressed

so as to fall into one of the cases treated. Rather than write out

w(x; e., a' , b, b I) everytime it is called for, we will often simply

write w Where the context makes clear what the contents of the

parentheses are.

3.2.2 The Maximum Absolute Difference Between Two Normal c.d.f. 's

Since K(z), the standard normal c.d.f., is strictly increas-

* *ing on (_00,00) and (x -Il)/rr = (x -Il')/rr' has a unique solution for

*x if and only if rr/= rr' , we see (by Theorems 3.3 and 3.4) that

any two normal c.d.f. 's with equal variances do not intersect any-

where, and any two normal c.d.f. 's with different variances inter-

sect once and only once.

For any value of (Il, Ill, rr, rr l ), w is differentiable with respect

to x for all x. Since

lim w=lim w=O,X-4 - 00 X -4 00

all absolute extreme values for w occur where dW/dx = O. Thus

in What is to follow, we locate those x-points where dW/dx = 0, and

choose from among those points x such thato

Case 2: H = H', cr <cr'

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••I

]

maxx

dW/dx = ° -+ x = ~ or Xq

If dW/dx = 0, then (X-'tJ.)2 = (x-'t!')2

If we assume cr< cr' ,

d [ 1 J(X-'tJ.)/cr J(X'1J.'Ycr'dW/dx = dX (~)- exp(-t2/2)dt - eXp(-t2/2)dt

-co -co

Case 1: H < hl', cr = crt

Thus

where

~,Xq = (a2-crt2)-1[a2'tJ.t-ert2'tJ.+ CTCT'.J('tJ. t _'t!)2+(a2-cr'2).tn((cr/cr')2} ].

We observe that x < x •q P

Let (x ,y ) be the point where w = 0. By Theorem 3.5, if x < x ,o 0 0

then w < 0, and if x < x, then °< w.o

Thus min loT = w(x ; 'tJ., 'tJ.' ,cr, crt)X q

max w = w(x ; 'tJ., 'tJ.', cr, cr').x p

Xp,Xq

= 't! :t CTCT,J(a2_cr'2)-1.tn{(cr/crt )2}

By Theorem 3.6, w(x ; 'tJ., 'tJ. ,cr, cr') = - w(x ; 'tJ., 'tJ., cr, cr') •p q

97

val. Hence if another point of intersection exists its x-coordinate

Hence max Iw I = "'l(x ;1l,Il, (J', (J" )x p

*The part of a

Applying Theorem 3.2, again we obtain

max Iwl = - min wx x

= w(x ; lJ., lJ.' , (J', (J" )q

J(x _lJ.' )/(J"

= ...L q exp(-~)dt

& (x -Il)/ (J'q

We can apply Theorem 3.2 in this situation.

is played by lJ.'. Thus we have

Case 3: lJ. < J..1', (J' < (J"

Case 4: H' < Hz (J' < (J"

If we say (J' < (J", then the c. d. f. 's intersect at one and only

one other point, (x ,y ), 0 < Y < 1. To see this, we observe that000

a value of x such that F(X;1,(J') = F(x;1',(J"), cannot be in the

interval (1',1 ] as F(X;1,(J') < F(X;1',(J") for all x in this inter-

3.2.3 The Maximum Absolute Difference Between Two Exponential c.d.f. 's

It is clear that any two exponential c.d.f. 's, F(x;1,(J') and

F(X;1 ' ,(J") coincide for x ~ min (1,1').

I

I.IIIIIIIIeIIIIIIII·I

It is clear that we may confine ourselves to the interval

[max( r, r' ) ,00) in a search for points where w attains extreme values,

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••I

r' < r ,

- r < - r' ,

r < x , as is shown by the following argument.o

that r < x • To say r < x is equivalent to sayingo 0

K«r-r')/o-') < K«x -r')/o-') =y. Thus we want to show that thereo 0

is one and only one point of intersection (x , y ) such thato 0

K«r-r' )/0-') < Y • To put the situation in the framework ofo

Theorems 3.3 and 3.4, let zL = (r-r')/o-', and let zu = 00 •

Now (x -r)/o- = (x -r')/o-' is satisfied by x = (o-'r - rTf')/ (0-'-0-) •000

must be in the interval (r, 00) • By Theorems 3.3 and 3. 4 we now show

that there is one and onl:y one point of intersection (x ,y ) sucho 0

If 0- < 0-', r < r' , then two exponential c. d. f. 's do not inter­

sect unless x ~ r, for if they did intersect, then by Theorem 3.4,

r < (ro-' - r'o-)/(o-' - 0-), but r ~ r' implies (ro-' - r'o-)/(o-' -0") ~ r •

Finally if 0- = 0-' , r f r', we see, again by Theorem 3.4,

that F(x;r,o-) f F(x;r' ,0-'), unless x ~ min(r,r') •

Also

Hence ZL = (r-r')/o-' < (xo-r')/o-' ,

and K(ZL) < K«xo-r')/o-') = Yo •

Thus by Theorems 3.3 and 3.4, for r < x, F(x;r,o-) and F(x;r',o-')

intersect if and only if x = (ro-' - r'o-)/(o-'-o-) •

Hence w attains a maximum for x = l' •

99

as if x < max(1,1'), w is a non-decreasing function of x. Further-

J (x-r')/lT'e-tdt Jo

maxx

z < 1', 0' = 0"

= (1/O')exp( - (x-1)/0') - (J./ 0" )exp( -(x-1')/ O'~ •

dW/dx = 0 -. x-1 = x-1', which has no solution in x.

Hence

1(1'-1)/0' t

Iwl = e- dto

= 1 - exp(-(1'-1)/0') •

When 0' f: 0" , dw/dx = 0 -. x = Xr '

Where ~ = (O'~') [ (J'(J"ln(O'/O") + (1'0' - 10")]

• exp [- (1-1')/(0"-0') ] •

( , ') [(0'/0'1 )0'/ (0" -0') _ (0'/0" )0" / (0" -0') ]and w x ;1,1 ,0',0' =r

Case 2: Z =7' I 0' < 0"

If we evaluate Wat max (1,1'), we obtain zero. Since

lim w = 0 , there must be a value of x in the interVal (1,00) suchx -'00that dW/dx = O. Such a value is provided by xr •

Thus

Case 1

mre, since w is differentiable in x except when x = max(1, 1') and

since lim W=0, we need only consider x such that x =max(1,1') orx -'00

such that dW/dx = O. (When we make statements about dW/dx we refer

only to x in (max(1,1'),oo).)

d [J (x-1/0' tNow dW/dx = ax 0 e- dt -

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100

unique value satisfying the equation

max Iw I = -min w(x; 1,1' ,0",0"')X

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••I

Iwl = max w(x;1, 1',0", 0"');

Iw I = -min w(x; 1, l' ,0",0"') •

*say 1, in the interVal (1' ,00),

Iw( 'V*''V* 'V' ~ ~') I That .fs, 'V* is the, " "'U'U. ...,

then maxx

then maxx

0" < 0'"

= 1 - [exp -(1-1')/0"'] •

If 1 < 1*, 0" < 0"', max Iwl = max w(x;1,1',0",O"').- x

*< 1 < 1 , 0" < 0"', then the only possible X-value at which

Hence for some value,

IW(Xr ;1*, 1', 0", 0"') I =

If l' < 1, (1': < 0'" , there are two x-points at which Iw I

may attain a ma.xinn.un, x and 1. For 1 in the interval (1' ,00) ,r

Iw(1;1,1' ,0",0"') I is a continuous function of 1, increasing from

zero to one and !vT(xr;1, 1',0",0"') I is a continuous function of 1

decreasing from [(0"/0"') 0"/(0"'-0")_ (O"/O",)O"'/(cr"O")] to zero.

By Theorem 3.2,

*if 1 < 1 , 0" < 0"',

*if 1 < 1, 0" < 0"',

*Case 3; l' < 1 S 1,

(3.1) [(0/0'" )0"/ (0'" -0") _(0"/0'" )0'" / (0'" -0")] exp [-(1*-1 , )/ (0'" -0")]

+ exp[-(1*-1')/0"'] - 1 = 0 •

If l'

max w can occur is where x = x. However if 1 < 1', 0" < 0"', thenx r

max w can occur either at x = x or at x = 1'. If l' < x , thenx r r

max w must occur at x , as d2w/af- < 0 for x = x • If' x < l' ,x ~ r r-there are no x-points in (.,' , co) such that dW/dx =O. That is,

I

I.IIIIIIII_IIIIIIII·I

w must be non-increasing in x for x in (,,' , co) as lim w = 0 •x -.co

Thus max w occurs, in this case, at ,,'. If 1ve observe thatx

,,' < x E - ) ,,' - cr .tn «(J"/cr) < 1,r

~ S'" E :." S'" - cr .tn «(J"/cr) ,

*we can sum up our results for " <" ,cr < crt , as follows.

*Case 4: z' - cr.tn «(J"/cr) < 1 < Z , (J' < (J"

max Iw I = w(~;1,,,' ,cr, crt )x

= [(cr/cr') cr/ (cr' -cr) _(cr/cr') crt / (cr' -(J')]

ae:x;p[-(z-Z')/«(J"-cr)] •

Case 5: Z < 2" - (J'.tn (cr'/ (J'), cr < (J"

max Iw I = w(,,'; 2', l' , cr, (J")

= 1 - e:x;p [-(,,'-1)/cr] •

101

Sub- section 3.2.1

102

Only results will be presented here. The distribution is

simple enough that the reader should have no trouble in verifying

I

••IIIIIII

_IIIIIIII

••I

From results previously obtained in this section, it fo1lo,,'sA

d = maxIF(x;e) - F(x;e ) I is parameter-free for all the cases- -0

max Iwl = 1 •x

max Iwl = 1.x

Case 1: H< H' , w = w'

them. They can be derived with the aid of the general theory in

Case 4: H' - Mwt-w') < H::::' H', w < w'

max Iwl = i - (~-~')/w' - ~/w' •x

max Iwl = (~' -~ )/w •x

Case 2: H' ~ H < H' + i{wt-w'), W < w'

maxlwl = i +(~-~'Yw' - ~/wr.x

Case 3: H' + i{wt-w') ~ H ' w < w'

considered in Sub-sections 2.4.1-2.4.3, 2.5.1-2.5.3, and 2.6.1-2.6.3.

By ~,we mean ({i, {i) or {i, or ~, where {i and ~ are the estimators,

that

Case 5: H~ H' - i(wt-w'),w < w'

3.2.4 The Maxi..mum. Absolute Difference Beti',een TYro Uniform c.d.f's

A3.2.5 The Distribution of maxIF(x;e)-F(x;e ) I in Certain Cases

x - -0

given in the appropriate sub-sections, of ao and bo respectively.

In every case, ~ and ~ are such that the distribution of

«~-a )/b ,~/b ) is parameter-free. Keeping this in mind, if we000

refer to the results in Sub-section 3.2. we can verify the assertion

I

I.IIIIIIII_IIIIIIII·I

103

made at the beginning of this paragraph. The verification for the

exponential distribution will be given here. In considering the

normal and uniform distributions, one would proceed along lines

similar to those used for the exponential. distribution.

In dealing with the exponential distribution, we first treat

the situation in Sub-section 2.5.1, where CTO

is known, but "0

is unknown and estimated by 'j. Let us substitute '1 , "0 and CTo '

respectively, in Case 1, SUb-section 3.2.3. We see that

d = 1 - exp{ -('1 - ., )/CT ) ,o 0

and is thus a function of ('1-.,o)/CTO)' a r.v. with a parameter-free

c.d.f. Bence the c.d.f. of d is also parameter-free.

Next we consider the situation in SUb-section 2.5.2, where

., is known, but CT is unknown and. estimated by ~. From Case 2,o . 0

SUb-section 3.2.3, we infer that

(3.2)

104

and is tlnls a f'lmction of a/&o' which has a parameter-free c. d. f.

Hence the c.d.f. of d is parameter-free.

Finally we consider the situation in Sub-section 2.5.3, wrere

both 7 and 0" are unknown and estimated by'1 and 'do respectively.o 0 , , ,

As before, we will proceed by using the results of Sub-section 3.2.3.

*We see that 1 is defined in Sub-section 3.2.3 as satisfying equa-A

tion (3.1). If we substitute 0" for 0"',0" for a, 7 for 7', and,0 0

'1 for 1* in equation (3.1), while assuming 'do < a , we obtaino

[('do/O" )'do/ (0"0-'d-t ('do/O" ) 0"j (0'0-'&1 exp [-(1-7 )/ (0" -'do)]o 0 ~ 0 0

-*'+ exp[-(7 -7 )/0' ] - 1 = 0 ,,. 0 0

Which defines 1'. Being defined as the solution of an equation in

which the random variable 'd- is involved, l' is also a random variable.

We should observe that the r. v., (:;-10)/0"0 is a function of 'd-/O'o.

This is evident if we rewrite equation (3.2) as

[('do/O' ) [(O"j'd-)-lrl

_ ('&/0' ) [l-td-/O"orl

] exp[-«T---1 )/0' }(l-'do,b: )-~o 0 _ 00,0

+ exp[-(T---1 )/0" ]-1 = 0 •o 0

(1*-10)/ 0"0 satisfies equation (3.3), which is expressed in terms of

the random variable '&/0"0. It is clear that 1* satisfying equationNIfo '

(3.2) is equivalent to (7 -10)/0'0 satis:f'ying equation (3.3). Thus,

since there is only one vaJ..ue of T-- that sa.tisfies equation (3.2) for

a given value of 'do, there is also only one value of (1*-70)/0"0 that

sa.tisfies equation (3.3) for a given value of 0"/0". This. 0

means equation (3.3) defines (1*-7 )/0" as ao 0

I

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_IIIIIIII

••I

I

I.IIIIIIII_IIIIIIII·I

10.5

function of 'd-/0"0. We will use this fact shortly.

-Let us also define x to correspond to x in Sub-sectionr r

That is, supposing 0" <'d-,o

Now, according to Sub-section 3.2.3, all values of (9',&'),

with the exception of those such that 'do = 0"0' fall in one of the

following 4 sets. (We need not consider any value of (~, 'd-) where

'd- = 0"0 ' as the set of a.l.l such instances has probability zero of

occurrence and hence the c.d.f. of d is unaffected by values of d

over this set) •

s - {(~ ,'d-) 10"0 < ~ , "I <~<x},A- o r

S = {(~,~) I~o < 0" , - 1\"I < x < "I} ,B o r-

Se = {(~ ,~) rd-< 0"0' - 1\"10

< 'Y* ~ "I} ,

Sn = {(~,'d-) ~ < 0"0' "I <~<7*}.0

We will consider each of the four sets above separately, and shOi'T

1) (~, 'd-) falling in any one of the four sets is equivalent

to «~-"I )/0" ,~/O" ) be:i,ng in a certain set defined independently of. 0 0 0

("10,0"0)' 2) given that ('1,'&.) has fallen into a particular one of the

four sets, d = rnaxIF(x;('1,'d-»-F(X;(7 ,0" »1 is a function of. :X .. 0 0 ..

«~-"I )/0" ,'d-/ 0" ). Once these points are demonstrated, it followso 0 .0

that the c.d.f. of d is parameter-free.

To help us in deIOOnstrating these points, 'We infer from Sub­

section ;.2.3 that

(3.4) 70 <'t< ~ ~ ~-O"o' J,n(&/O"o) < "10 ~ (~-70)/0"0 < In('d-/O''o)'

Thus

d = 1 - exp (- (~-7 )/a) •o 0

(~,{t)€ SA~ (1 < '0-/0" ,(':;-7 )/0" < J,n('o-/O"».o 0 0 0

I

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_IIIIIIII

·1I

106

We first consider SA. Using statement (3.4), we can see that

and

is in a certain set defined independently of (7 ,a ). AJ.soo 0

(~,{t)€ Sc means (7 ,~, a ,{t) comes under Case 3 of Sub-section 3.2.3.o 0

('1,'d-)€ R....~({t/ao < 1, 0 «~-7 )/a «~-ro)/a~---"'--«~-r )/a ,'d-/a)J) ooo~ 000

is in a certain set defined independently of (r0' ao). (':;,'0-) € SD

Also, since (~,'d-)€ SA means that (70'~'0"0,{t) is such as to cane

under Case 4 of Sub-section 3.2.3, we have (rewriting the appropriate

e:x;pression somewhat) that

d = [(0" /tr)rf&/O" )-lr1

_(0" ~)[l-O"o/~rl]e:x;p[_{(7 -'1)/0" ) «'0-/0" )_1)-1].o 0 0 0 0

d = 1 - exp(-('1- 7 )/a ) •o 0

In considering Sc ' we use the fact that (~-7 )/a is a func­o 0

tion of '0-/a , to validate the assertion thato

('1,'d-)€ Sc~('d-/qo < 1, 0 «1*-70

)/a,;s. ('1-r~a~(~-ro)/O"o,!d-/ao )

We next consider SB. Using statement (3.5), we can see that

(7,O")€ SB~ (1</\0"/0" ,tn(flJla)< (If-7 )/0"). Since (t:;,~)€ SBo 0- 0 0

means that (70'~'O"o''d-) falls under Case 5 of Sub-section ;.2.3,

we have that

In considering R...., we again use the fact, that (1*'-7 )/a is-D 0 0

a function of /(,/a0 ' in order to validate this time, tla t

I

I.IIIIIIII_IIIIIIII·I

107

also means that (1 /1,0" ,~) comes under Case 4 of' Sub-section 3.2.3.o 0

Rewriting the expression f'or max IwI there, we have thatx

d = [('d-/O" )[(O"j'd-)-lr1

_('d-/0" )[1-1t-/0"0rl

Jexp (_(c1_1 )/0" }(1-'d-/0" r 1) •o 0 000

Now we have denx>nstrated what we proposed, as regards the

four sets SA' SB' SC' and SD. We can conclude that the c.d.f. of

max IF(x;~,'d-)- F(x;')' ,0" ) I is parameter-f'ree.x 0 0

2.3. Expected Maximum Band Width

3.3.1 Preliminary Remarks

We now consider each of the nine bands constructed in Sub-

sections 2.4.1-2.4.3, 2.5.1-2.5.3, and 2.6.1-2.6.3. Let us modify

the bands pertaining to the exponential distribution, in that we

now assume that the location parameter, 1, can assume any real value.

Thus we avoid having to adjust parameter confidence regions for the

assUIIPtion that O.=:: 1, before we use them f'or band construction.

We will show that the ma.ximum. band width f'or each of the nine bands

we now have, does not change from sample to sample. Hence the

expected maximum band width, E(WM

), for each of' these bands is

simply the maximum band width for any given sample and can be

computed without great difficulty. Since E(WM) f'or a K-S confidence

band is simply 2D (where Pr(D < D ) = c), in many cases,n,c n - n,c

comparison of our nineparametric bands with the corresponding K-S

bands, as regards E(WM), can be conveniently made.

B falls under Case 1 of Sub-section 3.2.2.

From Sub-section 2.4.1, we have

I

.­IIIIIII

_IIIIIIII

••I

n-1)s2

: ~-1,-M1-C)

(n-1)s2

~-1,-M1+C)

B = [D{~), D{~)] ,

where ~ = x - e-M1+c)' (To/.[n , ~ = x + e!<l+c)' (To/.[n •

B={{x,y) l{x,y)€[D{~,(T2),D{~,(Tl)]' x~ x) u

(x,y) ,1(x,y)€[D(~,(T1),D(I-'2,(T2)~x < x) ,

Thus

3.3.2.3 Unknown Mean, Unknown Standard Deviation

From Sub-section 2.4.3, we have

3.3.2.2 Known Me~, Unknown standard Deviation

From Sub-section 2.4.2, we haV~

Where

3.32 Maximum Expected Width of Bands for the Normal Distribution

3.3.2.1 Unknown Mean. Known Standard Devie.tion

B falls under Case 2 of Sub-section 3.2.2.

If we let ~-l,-Ml+C)/~-l,-Ml-C)= r c ' we have

\t = ..l-J~ exp(-t2 /2)dt =E("M) •

~ J lnrcr -1c

109

,

,(n_1)s2

~-1,~1-c2)

Cf. =1

If we consider that part of B for Which x .:s x , vIe are dealing

vnth t,vo c.d.f.ls that fall under Case 4 of Sub-section 3.2.2.

Now we need only find E( max W(x;B», for W(x;B) is synnn.etricx<x

aroum 'i. That is, if (x:- x ) = -(~-x) , then

W(Xa;B) = W(~;B), or explicitly

To see this, we note that

if we prove that (Xa-~)/<T2 = - (~-~)/<T2 ,and

(Xa -~)/<Tl = - (~-~)<Tl. This is done below •

{Xa-~)/<T2 = [Xa-x + e-Ml+cl)<T2/.Jn]/<T2 = (Xa-X)/<T2+ e~l+cl)/-k •

where zl and z2 are any real numbers.

This means we will have Shovffi that W(x;B) is symmetric around x ,

I

I.IIIIIIII_IIIIIIII·I

110

(~-1J:3)/0'1 = [~-x + tM1+C1

)0'2/,[n]/0'1 = (~~X)/0'1+ ti(1+C1

)/..fu •

(~-~2)/0'2 =[(~-x .. t M1+C1

)02/,[n]/ 0'2 = (~-X)/0"2 - eM1+C1>'/~ •

Using the assumption that (xa-X) = -(~-X), we have

(xa-1l4)/0'2 = -(~-X)/0'2 +ei(1+C1

)/,[n = -(~-~2)/0'2 '

(xa- 1l1)/0'1 = 7(~-x)/0"1 - ti(J+c1)/,[n = -(~-~)/0'1 •

occurs) if x < x. We haveq-

x = (~-1') -1[~1l4-~I\ + 0'10'2.J (1l4-1l1)2+(c1.-~).tn {(0'1/0'2)2 )]fxomSub-sec%ion 3.2.2.

Now ~1l4-~1\ = ~(x - eM1+c1).O'i~ ) - ~(x + tM1+c1)0'1/~)

= x (of -~) -(eMl+C1

)/,[n110'2(0'1+0'2) ,

and (1\-~1)2 = [x - tM1+c1

)O'i,[n - x - ti(l+C1

)0'1/,[n:F

= e2-M1+c1) [0"2+0"1:FIn.

Hence

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_IIIIIIII

·1I

3.3.' Maximum EJcp::cted Width of Bands for the Exponential Distribution

3.3.3.lUnknown Initial Point, Known Standard Deviation

1ll

,

(Xq-J.l4)/(]'2 = (K1;-Kr)-l[¥-M1+cl,l.frI

- l1, In-~~1+Cl)+(KU-~)(Ku+Ki)-l.tn((Ku/~)2) ] ,

(Xq-~)/(]'1 = (Ku-Kr) -1 [11.t -Ml +cl)/..fn

-Ku- In-1 t~l+Cl) ~-(KijI<I) (~+I1);I;;{(Iu7K~:>2)] ,

we have xq < x .Thus we can say

J(x -~)/(]'

WM

= ~ q 2exp (_t2 /2)dt •

21t (Xq-~)/(]'l

If we let XU = 1 , l1, = 1.J~ JJ) .Jv2 M )"n-l,2\l-C2 "n-l,2\1+C2

we obtain, after some algebraic manipulation,

B = [D(rl ),D(r2)], where 7 1 = X(l)-~,-Ml+C)(]'l/2n ,

' 2 = X(1)-~,-M1_C)(]'1/2n •

From Sub-section 2.5.1,

and we see that

This falls under Case 1, Sub-section 3.2.3.

I

I.IIIIIIII_IIIIIII

••I

112

3.3.3.3 Unknown Initial Point. Unknown Standard Deviation

From Sub-section 2.5.3 ,

I

.'IIIIIII

_IIIIIIII

••I

B = [D(0'1),D(0'2)] ,n

0'1 =2 i~l (xi-X(1»/~-2,-M1+C) ,where

This falls under Case 2, Sub-section 3.2.3.

From SUb-section 2.5.2,

3.3.3.2 Known Initial Point1 Unknown Standard Deviation

B = ((x,y) l(x,Y)€[D('12, 0'2),D('14' 0'1)]' x:s x(l)}

u((x,y) l(x,y)€[D('1l 'O'l),D('13'0'2)]' x(l) < xl •. n

'11 =x(l) - ~,i<1+C1)0'1/2n, CT.l = 2 i~ (xi-X(1»/~n-2,i<1+C2)'

n'12 = X(l)-~,~1+C1)0'212n , CT2 = 2i~(xi-X(1»/~-2,~l-c2)'

'13 =x(l) -~,~1_Cl)0'2/2n ,

'14 = x(l) -~,i<l-cl)O'J2n •

We note that '12 < '14' 0'1 < 0'2 •. Hence F(x; ('12' 0'2» and

F(X;('14'0'~» intersect at a point (xo'Yo). In SUb-section 2.5.3,

it was shown that F(x; ('14' 0'1» :s F(x;('12,CT2» for x:s x(l). Thus

0'11 ( 0'2-0'1) 0'21 ( 0'2-0'1)Thus WM= (0'1/ 0'2) - (0'1/ 0'2) •

If we let f c = ,~-2,i<1+c)/~-2,~1-C) ,

Then W = f (l-fc)-l_ f fc~-fc)= E(~) ,M c c M •

ll3

x > x(l) , than it has at x(l) , it is clear that W(X;B) attains its

maximum at x = '14 •

(xo'Yo) f. f(x,y) lex:,y)€ {[D('Y2'0'2),D{'Y4'0'1)],x,sx(1)J, and hence an:l x-point

"There Iw(x; '14' 1'2' ()l' 0'2) I may possibly attain a maximum by virtue of

the fact that Q\'T(x;'Y4,'Y2'0'1'0'2 )/dx =0 for that point has a

value greater than x(l)' This means that the only x-point not larger

than x(l) such that IW(x;B) I may possibly be ma.ximu.m, is x = '14'

Now we consider the portion of B for which x > x(l)' We can

describe this portion of B as ((x,y) I(x,y) [D(11'0'1),D('Y3

, 0'2)]' x(l.f x}.

Also W(x;B) =Vl(X;'Y1 ,'Yj,cr1 '0'2)' x(l)< x •

Hence if w(x; '11, '13' 0"1' 0"2) does not have a larger value for some

dw(x;11, '13' 0'1' 0'2)Suppose dx ~ 0 for x =x(l) •

Since lim w(x;'Y1,'Y3'0"1'0"2) = 0, and w(x;''Yl ,'Y3'0"1'0"2) isx -+00

differentiable in x for x ~ x(l)' if for same x > x(l)'

w(x;'Yl ,'Y3'0'1'0"2) > W(X(1);'Yl ,'Y3'0'1'0'2)' then

dw(x;'Y1,'Y2'0"1'0'2) =0 for at least two distinct x-points. But thisdx

cannot be, as accordirig to Sub-section 2.5.3, the solution of

dw(x;'Y1,'Y3'0"1'0"2)On the other hand, suppose dx > 0 at x =x(l)'

This means sup "I'~ (x ;'11,'13'0"1,0"2) ~w(11);'Y1''Y3'0"1'0"2)' Sincex(l)<x

:l-im w(x;'Yl ,'Y3'0'1'0"2) = 0, w(x;'Yl ,'Y3'0"1'0'2) attains this supremumx-+oo

at some x-point, and w(x; 1'1,1'3,0"1' 0"2) being differentiable for

I

I.IIIIIII,I_IIIIIII,.I

114

dW(X; "1' "" 0"1' 0"2) =0 at this point, which must bedx

the point x eXhibited in Sub-section 2.5.3.r dW{x; "1' "3' a: ,0"2)

We see that if we can show that the sign of dx 1

at x =x{l) is independent of the particular sa.nq>le values present,

and that the value of W{x;B) at r4 and xr is independent of the

particular sample values present, it will follow that WM

is constant

from one samole to another.

dw(x;"1'''3'0"1'0"2) d [(x-r,) (x-rl ) ]:New dx c: dx exp{ - ) - exp{- )

0"2 0"1

dw{x;rl ,r3'0"1'0"2)Let dx > 0, at x = X{l). This means

In 0"1 + (x{1)-r1)/0"1 < In 0"2+ (x{1)-r,)/0"2)'

(X(l)-"l)/O"l - {x(1)-r3)/0"2 < Jn0"2 - tnO"l •

Making the proper SUbstitutions, we find that

(X{l'-"lYO"l~-~,l<l+Cl), (x(1)-r')/0"2 =~-l~,~l-cl) •

Thus (X(1)-r1)/0"1 - (X{1)-r3)/0"2=in-l[~,i<1+cl)-~,i<1-cl)] •

Also ln0"2- mO"l = In(O"!O"l) = x2n-2,i(1+c2) •

~-2,-MI-C~

Hence

] .

II

.11IIIIIII,I

_IIIIIIII

••I

1 ,~-2,-Ml+C2)

Next let K =a

dw(x;rl,r3'~1'~2)For = 0, or < 0, a corresponding result holds by a

dxdw(x;rl,r3'~1'~2)

similar argument. Thus vTe see that the sign of dx

is ind,e.pendent of the particular sample values present.

115

• exp [-(rl-r3),(~2-~1)]

K /(K -K ) K (K -K )= [(Ka!~) a; -0 a -(Ka/RbfO -0 a]

which is independent of the particular sample values present.

Also W(Xr;B) = w(Xr;rl,r3'~1'~2)' and from Sub-section 3.2.3, we

see that

We can conc,lude that the value of WM is constant from one

sample to another.

.exp[-in-l(~ JJl-c )I<b-~ JJl+c )Ka)/ (Rb-Ka )] ,,2\ 1 ,2\ 1

which is also independent of the particular sample values present.

I

I.IIIIIIII_IIIIIII,.I

~.3.4.3 Unknown Mean, Unknown Range

From StiQ-section 2.6.1,

From Sub-section 2.6.3,

II

.IiIIIIIIII

_IIIIIIII.,I

,

B = [D(tA).),D(w2)] ,

~ = r/R~l+C)

w2 = r/R~l_C)

where

B = [D('"1.,~),D(~,w2») ,-1

where '"1.::: m -tir(l-R[c]-1), tA). = ~(l+R[c] ) ,

~ =m +kr(R-1 -1)[c]

ll6

Thus WM ::: -M1-~/~)

=l(l-R~l_C)/R~l+C)] =E(WM) •

This falls under Case 2, Sub-section 3.2.4.

B = [D('"1.),D(~)] ,

where '"1. = m - M~l+C)~ ,

~ = m + M~l+C)tA). •

This falls under Case 1, Sub-section 3.2.4.~-'"1.

Thus WM= tA).

3.3.• 4.2 Known Mean, Unknown Range

From Sub-section 2.6.2,

~4 Maximum Expected Width of Bands for the Uniform Distribution

J.3.4.1 Unknown Mean, Known Range

II,eIIIIIIII_IIIIIII.eI

ll7

This falls under Case 1, Sub-section 3.2.4.!J,2-U.Thus W

M= . J.

UJ.

3.3.5 Invariant Properties of Certain Confidence Bands

We have seen that in many cases the ma.xinn.ml band width is the

same regardless of the sample values. In this sub-section we explore

why this is so and find in the process that the bands considered

previously in this section also have other properties that one in-

variant under sa.m;p1ing. The underlying assumptions and notation here

will be the same as for the general theory in Sub-section 3.2.1.

Lemma. 3.7 If m is a sub-set of ((a,b) \_00< a < 00, 0 < b) ,

then M(m) = ((x,y) \x = a+bz, y =K(z), -ClO < z < 00, (a, b)€ m) •

Proof: By the definition of M(m) in Section 1, we have

M(m) = ((x,y) I y = F(x;(a,b», (a,b)€ m) •

If y = F(x; (a,b» as stated at the beginning of Section 2,

(a,b)€ m, then under our assumptions on F(x;(a,b» as stated at the

beginning of Section 2, y == K(z), where x = a+bz, (a, b)€ m•If x = a + bZ, Y = K(z), (a,b)€ m, then z ~ (x-a)/b, so that

y = K«x-a! b) .F(x;(a,b», (a,b)€ m•

Theorem :2.8 Let m be a sub-set of

((a,b)\- 00 < a < 00,0 < b) = {(a,b)} say.

Map {(a,b)} into {(a',b'»!- 00 < a'<oo,O<b'}== {(a',b'») say,

li8

by means of the transformation a' = ~+c~, b' = cTb , where ~

and cT are fixed numbers such that - co < ~< co , 0 < cT. Let m' in

{(a' , b ' )} be the image of m. Let H(m ') be the matting based on m' ,

where y' =F(x';(a',b'». (Thus H(m') is a sub-set of the

(x' ,y')-set.)

Then H(m') = {(x',Y')I x' = a.:r+c~, Y' = y,(x,y)€ H(m)}.

~: Since by Lemma 3.7,

M(m') = {(x',Y')lx' =a'+b'z,y' =K(z),- co<z<co,(a',b')e m'} ,

we want to show that

{(x',y') lx' = a'+b·Z,Y' = K(z),-co<z < co,(a',b')€m')

= ((x',Y') lx' = a.r+c~,Y'=y,(x,y)€ M(m)} •

If' (x', Y') is in the left-hand set above, substituting for

a' and b t, we have x' = ~+c~+cTbz = ~+cT(a+bz), (a,b)e m. If we

put x = a + bz, Y = K(z), then (x,y) e M(m) by Lemma 3.7. Thus if

(x' ,y') is in the left-hand set in the above equation, it is also in

the right hand set.

If .(x' ,y') is in the right-hand set above, then

x' =~+ cT(a+bz), y' =K(z),(a,b)€ m, by Lemma 3.7, or

x' = (~+ cTa) + cTbZ = a' + b'z,. (a',b')€ m', by our assumptions.

Thus (x' ,y')e M(m') by Lemma 3.7.

Theorem 3.9 Under the conditions of Theorem 3.8,

(yl(xo'y)€ M(m)} = (y' I(~+ c~o,yt)€M(m'»

for all x on the real line.o

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Proof:-

= (y Iy = K{(x -a)/b), (a, b)€ m} •o

(y' I(~+cr 0' y ') €M(m f») = {yt 1:1 '= F(~+c~0; (a ' , b ' ) ); (a', b ' ) € m' }

= (yfI7 = K«~+c~o-af)/b'); (a',b')em')

.. (y°IY'= K( (~+c~o -~-c.ra)/ cTb), (a, b) em}

= (y1 y'= K«x -a)/b), (a,b)€ m} •o

Since the tl'lO sets given in the statement of the theorem equal

the same set, they equal each other.

Corollary 3.10 Under the conditions of T11eorem 3.9, if

M(m) is a band, then W(x ;M(m» = W(a-.+c-x ;M(m'» for any x on theo 'J: 'J: 0 0

real line.

Proof: First we should remark that it is legitimate to

speak of W(B.rr+cTxo;M(m'» as if M(m) is a band, M(m') is also a band

as is apparent from Theorem 3.9.

Corollary 3.10 itself is also apparent from Theorem 3.9, as if

(Y1(xo,Y)€M(m)} == (yf 1(a...r+cro'y') €M(m')} , where both sets are

intervals, then the lengths W(xo;M(m» and w(a...r+Cro;M(m'», of the

sets are e<lual.

Let q be the graph of W( x; M(m ),and let q.' be the graph of

WQc' ;M(m f». It is possible to set up the one-to-one correspondence

(x,y)~ (a..r+c~,y)

between q and q', where the left-hand point is in q, and the right­

hand point is in q'. From the fact that this correspondence

]20

preserves the value of the ordinate, it immediately follows that the

range of W(x';M(m'»:is the same as the range of W(x;M(m». Hence

max W(x;M(m» = max W(x';M(m'». Also it is apparent that the one-to-x x

one correspondence preserves the order of the obscissas. Thus we may

say that the order in "'hieb the values of the common range are plotted

against the obscissa does not change from W(x;M(m» to W(x' ;M(m'»

Each of the nine confidence bands, discussed previously in

this section, is the matting based on a chance selected region for

(a ,b) from a fa.mi.1.y such that any two regions, mand m', in theo 0

family are related by a transformation of the form

a' = a.r+cTa , b I = CTb ,

where mt in (a',b'») is the image of min (a,b}) under the trans-

formation.

To see that this is so, refer to the fifth part of SUb-sections

2.4.1-2.4.3, 2.5.1-2.5.3, and 2.6.1-2.6.3 Where the pertinent families

of confidence regions are described. (It should be remembered that

the confidence regions in Sub-sections 2.6.1 and 2.6.3 are to be

amended in accordance with the assumption that the location parameter,

r , can assume any real value.) One should then note that a typical

member of each of these families can be expressed as all points in

(a, b») satisfying a system of inequalities involving ~ and ~ ,

." A.mere a and b are the estimators, given in the appropriate sub-

sections,of a and b respectively. For those families where b is000

known, we set ~ equal to b. For those families where a is known,.00

we set ~ equal to ao•

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members of the range occur when plotted against the x-axis.

location parameter is plotted.

•b' = c aT

region undergoes an expansion or contraction by a factor constant

The reader can verify that in each of the nine families of confidence

regions under scrutiny here, if

confidence regions being discussed, satisfy the conditions of

Theorems 3.8 and 3.9. Hence we may deduce that WM is invariant under

sampling, as we have already seen for each of the nine bands examined

Thus any two members, mand !R', in any one of the nine families of

Now suppose that m, a member of one of the nine families of

confidence regions being considered, is described by a system of

ineClualities in 'd and 13, and that m', another member of the same

family, is described by a system of ineClualties in 'd' and ~'.

A, JlI. 1\, AcT = 15 I 0 , ~ = a - cTa ,

then m' in «a I,b')} is the image of min (a,b)} under the trans­

formation

in this section. Also, for each of the nine bands, the range of

W(x;B) is invariant under sampling along with the order in which

for all points, after whiCh the confidence region is translated a

certain distance in the direction of the axis along which the

Geometrically speaking, we see that we can expect a confidence

band to have these invariant properties under sampling, when the

confidence region for the location and scale paraIOOters on which it

is based, changes from one sample to another, only in that the

radius vector from the origin (0,0) to every point in the confidence

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3.3.6 Numerical Re'suJ.ts on Expected Maximum Band Width for the

Normal. Distribution

Table 3.1 will be discussed here. The computations far the

table are explained in the appendix.

First we will review how the K-S band is farmad. The K-S band,

with confidence coefficient c, is, as stated in the introduction,

based on the D -statistic, as to form the band, we need to know thatn

value D , such that Pr(D < D ) = c. Also 1et S (x) be the~c n- ~c ,n

sample distribution function for a sample of size n. That is,

Sn(x) =0, x < x(l) ,

Sn(x) = iln , xCi) ~ x < x(i+l) ,

S (x) = 1, x() < x ,n n -

where the xCi) are the order-statistics, x(l)~ x(2)~ ... ~ x(n). Far ,

any x-value, consider the interval [max(O,S (x)-D ),min(l,S (x)+D »).n n,c n n,c

The set of all such intervals, _00 < x < 00, 'constitutes the K-S

band on the c. d. f. of the population we are sampling from. Provided

that [S (x)-D ,S (x)+D J C [O,lJ for some value of S (x) , then n,c n n,c n

ma.xinmm band width for every sample, and hence the expected maximum

band width over all S8.JliUes is 2D • In Table 3.1, c = .95, andn,c

since [S (x)-D '95' S (x)+D 95 J C [O,lJ for some value of S (x)n n,. n n,. n

in every K-S band shown, the expected ma.ximum band width for all the

K-S bands in Table 3.1 is 2D 95.n,.The parametric confidence bands to 'Which Table 3.1 pertains,

are those described in Sub-sections 2.4.1, 2.4.2, and 2.4.3, where

a normal r.v. is treated. The confidence coefficient, c, was set

equal to .95 for the parametric bands for which o~ one parameter

was unknown. However the band of Sub-section 2.4.3, for 'Which both

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123

1J.0

and 0"0 are unknmm, is nat f'ttlly speci'fied given that c = .95.

We still have to decide what cl and c2 should be, sUbject to the

requirement that .c = cl c2 = .95. It was 'found convenient to put

cl = .9694, c2 = .98, in which case, c = cl c2 = .950012, which

would seem close enough to .95 for our purposes.

For each sample size n, the bands in decreasing order of

expected maximum band width are the K-S band, the parametric band with

both IJ. and 0" unknown, the parametric band with on1y~ unknown, and000

the parametric band with only 'b unknown. For all bands tabulated,

both the e:x;pected maximum band width and the rate of decrease of the

expected maximum band width decrease as n increases. For all three

parametric bands studied, the ratio of the expected maximum para-

metric band width to the expected maximum K-S band width would seem

to go to an asynptotic value, as n increases, nat markedly different

from the values the ratio assumes when n is relatively small. For

the band where only IJ. and 0" is unknown, and for the band Where onlyo 0

\l is unknown, this ratio appears to be an increasing function of n,o

whereas for the band where only O"ais unknovm, the ratio is

apparently a decreasing function of n.

124

Table 3.1 Expected ma.x:inIu:rn width (E{WM» of .95 confidencebands for a normal distribution

Col. (1) Parametric Band Parametric Band ParametMgd

K.s Band ~o Unknown ~o tJnknown ~o Known

cr Unknown cr Known cr Unknown0 0 0

Sample E(WM

) E{WM

) E{WM) as a E(WM

) E{WM)as a E{WM

) E{WM)af,f

10 of Col. (1) 10 of Col. (1 a %ofSizeI Col. (1'

10 .8185 .6004 73.35 .4646 56.76 .2191 26.77

15 .6752 .• 5071 75.10 : .3872 57.35 .1772 26.24\

.5882 .4466 I .3388 57.6020 ! 75.93 .1525 25.9324 I .5386 .4112 76.35 .3109 57.72 .1388 25.77j

31 l .4758 .• 3654 76.80 .2752 57.84 .1217 25.• 58I'

.4152 77.14 .2405 .1056 25.4341 I .3203 57.92I

51 ! .3732 .2886 77.33 .2163 57.96 .0945 25.32I

61 i .3418 .2648 77.47 .1982 * .0864 25.• 2857.99

71I .3173 .• 2460 77.53 1.1839 57.96 .0Boo 25.21I

~

81 " .2974 .• 2308 77.61 i .1724 57.97 .0748 25.15f91 .2808 j.2180 77.64 !.1628 57.98 .0706 25.14t i

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125

3.4 Expected Width of Confidence Bands at Selected:points

If we sup1?ose we are sampling from a population with known

c.d.f., F(z), and construct on the basis of our sample, a certain

confidence band, B, on F(z), then for any specified value, z say,o

we may be interested in the expected width of the confidence band B

at z. That is (using notation developed at the beginning of thiso

chapter), we may vlant to know what E(W(z;B» is for any value, z , of. . 0

z. If F(z) = K(z), the known continuous c.d-f. postulated at the

beginning of Section 2, knowledge of E(W(z;B» may aid us in assess­

ing the merit of B as applied to the c.d.f. of F(x;e ), Where e is-0 -0

a vector of unknov1ll parameter values, and F(X;~) is related to K(z)

as described in Section 2.

We will now derive an expression for E(W(z;~», where

F(z) = K(z) is as given in Sub-section 2.4.3, and BN is the confi­

denceband, M(m), described in part 9 of Sub-section 2.4.3. Thus

in this example, F(z) is the c.d.f. oft, a N(O,l) r.v. In the

following discussion, we will use the notation of Sub-section 2.4.3,

except that instead of denoting a a value of tl1e r.v. involved by

x, we will denote it by z.

Referring to SUb-section 2.4.3, we see that if z is a partie­o

ular value of z, then

z < Z0- ,

z < zo

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•

z < z ,0-

z < z ,0-

s

~-+, -M l - c2 )

n(n-l) .=

= f ~-l, -MI - C2)

I\J (n-I)

Pr(z < 'Z) = Pr(.[n z < e) ,0- 0-

Pr(z < z )= Fr(e <.[n z ).o 0

We note that

and

126

the random variable with this truncated distribution by

T- 1 r • Then we can sayn- ,'\In z

o ~-l, ~1-c2) _(zo-~4)/<T2 = Tn _I ~ z +

n(n-l) '0

Since e is a N(O,l) r.v., - ~ z is as N(O,l) r.v., and s is

distributed as J(~_l)/n-I. Thus «~ zo- ~ z)/s) has the non­

central t-distribution with (n-l) degrees of freedom and non­

centrality-parameter .[n zo. If we impose the restriction, zo:s z ,

then «J'n z - ~ z)/s)haS this non-central t-distribution,- 0

truncated so as to include only non-positive values. Let us denote

Also

We have now developed expressions for the terms in

Evaluati.ng this form for E(W(Zo;l\i» by means of numerical

integration is discussed in the appendix in the discussion of the

127

•

•

,-1,-Ml +c

2 ) +-------;.... T .Jr +

n(n-l) n-l, n Zo

~-l, -Ml-c2)

n(n-l)

(z -IJ-)/a: =o .;) 1

Similarly we can say...-=------.;:. 1 Yl+c ) elJ

21 l+C

l)"n-,~ 2 2\

(zO-~)/al = /----.;;;... T- .Jr -n(n-l) n-l, n Zo ~

Now ("[n z -..r;." i)/s , given that i < z , has a non-central t-o 0

distribution with (n-l) degrees of freedom and non-centrality para-

meter ..[n z , truncated so as to include only positive values. Weo

will denote the random variable with this truncated distribution by

T~_l,..[n zo· Some algebraic manipulations will reveal that

t-M l +cl)

ok.

computations for Table 3.2.

We will next derive an expression for E{W(Zo;~_S»' 'Where

B.._ is the K-S band with confidence coefficient c on any knownn:-~

c.d.f. F(z). From the definition of the sample distribution

E(W(Zo;~»' which if substituted in our original formula. for

E(W(zo;~»' yield a form amenable to calculation.

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128

in [O,n].

function given in Sub-section 3.3.4, we observe that for any fixed

Now consider the following two statements.

i iStatement I : - - D < 0 and - + D >1, where i is an integern n,c n n,c

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S (z )-D < 0 and S (z )+D < 1 ,n 0 n,c n 0 n,c-

S (z )-D < 0 and S (z )+D > 1 ,n 0 n,c n 0 n,c

S (z )-D > 0 and S (z )+D < 1,n 0 n,c - n 0 n,c-

S (z )-D > 0 and S (z )+D > 1.n 0 n,c - n 0 n,c

,,

= l+D -S (z ),n,c n 0

= 1

=2Dn,c

W(ZO;R__S) =D +S (z ),-.I{ n,c n 0

Let ~ be the greatest integer in [O,n] such that

~ - D > 0 and ~ + D < 1.n n,c - n n,c -

point, Zo say, the sample distribution function at the patnt,

Sn(zo)' is a random. variable whose value depends on the particular

sample values present. W(zo;~_s)' as a function of the random

variable, Sn(zo)' is defined as follows.

satisfies Statement II, and vice-versa. Also D may be such thatn,c

no integer satisfies either Statement I or Statement II. Thus Dn,c

can fall into one of three mutually exclusive and exhaustive cate-

Statenent I implies 2D >1, whereas Statement II impliesn,c

2D < 1. Since both statements cannot be simultaneously true,n,c -

it follows that if any integer satisfies Statement I, no integer

in [O,n].

Statement n: J. - D > 0 and J. + D < 1, where j is an integern n,c- n n,c-

gories. We will develop an expression for E(W(zo;~_S» where

D is such that Statement II is satisfied. If D is in one ofn,c n,c

the other two categories, a similar expression can be developed.

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129

Let ~ be the greatest integer in [O,n] such that

~-D >0 and ~+D <1.n n,c - n n,c -

Then from the definition of S (z ) and W(z ;BK s), we haven 0 0-

~E(W(Zo;~_S»= k:C(~HF(Zo)]k[l-F(Zo)]n-k [Dn,c + *]

kn ] ,

To simplify the above expression, we should note that

n. 1: (kn) [F(z ) ]k[l_F(z ) ]n-k = [F(z )+(l-F(z »]n = 1 •k=O 0 0 0 0

For :further simplification, we can use Gruder' s Formula [Johnson,

1957], which is

n (n) k n-k( ) (n) m n-m+11: k P q k-np = n m p q ,k=m.

where (p+q) = 1. Applying Gruder' s Formula in the form

n n~ k (n) k n-k (n) m n-m+1 ~ (n) k n-kLo k pq =mm pq +np Lo k Pq ,k~ k~

we find that

n- I: k(kn) [F(z ) ]k[l_F(Z )]n-k)k-n +1 0 0- 1

n= F(z ) I: (kn)[F(zO)]k[l-F(Z )]n-k

o k=O 0

(n +1) n +1 n-n-!-- ( n+1)[F(Z )] l[l_F(z)] 1

n n1

0 0

and that

n+ F(z ) I: (kn)[F(Z) ]k[l_F(z ) ]n-k

o k=~+l 0 0

Thus we obtain tha.t

130

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131

In evaluating the above expression, use can be made of tables of the

binomial distribution such as National Bureau of Standards, [1950].

Table 3.2 compares E(W(z;BK_S» and E(W(z;~» at selected

points, where F(z) is the c.d.f. of a N(O,l) r.v. and n = 24. Each

of the selected points is specified by the probability that a N(O,l)

r. v. does not exceed it in value. The computations for Table 3.2

are discussed in the appendix.

Table 3.2. Expected width at selected points (E(W(z;~ g» andE(W(Z;~T») of .95 confidence bands for a standardnormaJ. tIistribution based on a sample of 3ize 24

lPr(e, ~ z) E(W(z;~_s» E(W(Z;~)) E(W(Z;~»

E(W(z;l1c_s) )

.5 ,.54 .37 ,.69.• 25 .49 .08 ,.16

.15 ,.42 .07 ,.17

.10 ,.37 .07 .19

.04 ,.31 .05 .16

.01 .28 .03 .11

132

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133

3.5 other Criteria

As distinct from the expected mxi.mum width of a confidence

band on a c.d.f., y = F(z), we can conceive of the maximum

expected width of a confidence band. This measure of goodness of

a band would be the maximum value the function of x, E(W(x;B),

achieves over a.ll x.

We may set ourselves the goal of finding bands, sUbject to

certain restrictions, that minimize the expected maximum width or

maximum expected width.

Under certain conditions, we may be able to consider the

confidence band on a c. d. f., as a set of simulataneous confidence

intervals on the percentage points of a distribution. In that

case, we would be interested in criteria for the horizontal dimen-

sion of a confidence band analogous to the criteria we have for

the vertical dimension of a band. In order to be more precise, ,'re

make the following definition.

If B is a confidence band, y is a fixed value of y, and tIleo

x-set (x, y ) I(x, y ) € B} is an interval, then the length of theo 0

interval (x,y ) I(x,y )€ B} is the breadth of thecmfidence band, B,o 0

at the point Yo •

Criteria involving the concept of breadth are the expected

breadth of a band at a certain point or the e~ected maximum breadth

of a band. To illustrate matters here, let us apply the criteria of

expected maximum breadth to the K-S band. We observe that if

y < D , or if l-D < Y , the breadth of the K-S band at theo n,c n,c 0

point y is infinite, regardless of the particular sample valueso

present. Hence the expected maximum breadth of the K-S band is

infinite.

Another concept that might be used in setting up criteria is

band area. It is clear that the K-S band has an infinite area. As

for the parametric bands, we will now show that each of the nine

parametric bands treated in Sub-sections 3.3.1, 3.3.2, and 3.3.3,

has a finite area for any given s8JJi>le. There is no difficulty in

seeing that each of the bands for the uniform distribution, treated

in Sub-section 3.3.3 has a finite area. But it might not be obvious

that the bands for the normal and exponential distribution, dis-

cussed in Sub-sections '.3.1 and 3.3.2, also have finite areas.

Tl1e problem of' showing this for these other bands comes down to

showing that the area enclosed between any two normal c. d. f. 's or

any t'WO exponential c.d.f. 's is finite. Tha.t is, if F(t;~l) and

F(t;~) are the t'WO normal or two eJq>onential c.d.f. 's involved,

,.,e want to show that

is convergent. This 1'1ill follow, if we can shovl that

f eo [l-F(t; e) )dtx - to 1

is convergent uhere F(t; e) = - f exp( -(s-IJ.)2/2a2)ds and- .]21Ccr ~ eo t

- 60 < X < eo , or where F(t;e) = - f exp(-(s-7)/cr) ds ando - cr 7

7 < xO• And this in its turn can be proved by direct application of

the following theorem, [Taylor, 1955, p.653].

Theorem Consider an improper integral of first kind of the form

feo +(u)g(u)du ,Xo

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where the functions + and g satisfy the conditions:

b) g(u) is continuous and the integral

J ~g(U)dU is bounded for all xl > x •x - 0o .Then the integraJ. f 00 <Il(u)g(u)du is convergent.

Xo

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a) +'(u) is continuous, +'(u) < 0, and lim +(u) = 0 ,- U -+00

135

4. EXPLORATION OF FURTHER TOPICS

4.1 Confidence Bands on Selected Cumulative Distribution Function

Values and Percentiles of a Normal Random Variable

4 .1.1 Preliminary Remarks

SCheffe'-type simultaneous confidence intervals far linear

functions of the parameters in the general linear nx:>del [Scheffe',1959]

have at tim:!s been found too large, and attempts have been made to

find intervals that would be shorter over a particular subset of

linear functions of the parameters in which the investigator is

especially interested. For example, Tukey [Scheffe~ 1959] treated the

case where confidence intervals on all possible differences between

treatment means are desired, and Dunnett [1955] has dealt with the

situation where one is interested in the differences between the

mean for a control treatment and the means for all other treatments.

Also work has been done on finding confidence intervals for the

ordinates of a regression line over a pre-selected set of absQissa

points [Gafarian, 1963], and on finding confidence intervals for the

ordinates of a response surface over certain subsets of the independ­

ent variables involved [BOhrer, 1964].

Problems similar to those described above can be posed within

the context of the present thesis. That is, one may be concerned

mainly with the value of the true c.d.f. at particular points and

desire a greater degree of accuracy in knowledge of these points

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137

coofidence bands on percentiles or values of the c. d. f. of a norml

is distributed as T 1 r U h ren- ,''In Ko ' w e

, the true value of UK. It follows from this fact

Let T 1 A denote a non-central t-random variablen- ,

degrees of freedom and non-centrality parameter A •

y =F(x;(llo ,ITo»' over all values of x in the interval [xA'XB].

Suppose a sample of size n is taken fran a N( Il ,02) population.o 0

with (n-l)

is available for forming a confidence interval on the particular

such problems, but will confine discussion to methods for constructing

In this section we will not attenpt exhaustiveness in handling

distribution may be of special interest.

than of other points or particular percentiles of the underlying

is appropriate wilen no statistic with a parameter-free distribution

rationale behind it is given in lobod [-1950, p.229 J. The procedure

confidence intervals for the set of values of a normal c.d.f.,

We will suggest one way of constructing a set of simultaneous

r.v. over specified intervals.

on the fact that

(X1Cllo)

Also let T 1 A be that vaJ..ue of T 1 A such thatn- , ,P n- ,

Pr(Tn_l,~ Tn_l,A,P) =p. Johnson and Welch [1939] indicate how

the non-central t-distribution may be applied to obtain a confidence(JX-Il)

interval on a parametric function of the form IT = UK ' where ~

is a fixed value of x. The general procedure they use and the

ation of concern to us here can rest, according to Johnson and Welch,

4.1.2 A Confidence Band for a Normal c.d.f. on a Selected Interval

parametric function of interest. Use of the procedure in the situ-

I

I.IIIIIIII_IIIIIII••I

ment of the result that follows is essentially reproduced from Wilks

Now we define the parametric transformation,

138

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I'_I

IIIIIII

·1I

(the true value of UA) ,

"B-~u = - •B (J"

U~-~ LK

F(~;(~,O"» = F( 0" ;(0,1» = F(UK;(O,l» = ~ exp(-t2 /2)dt

We will use Tukey' s result to put simultaneous confidence intervals Ql

xA-~o

UAO = ~

then ['1a.' '1<:2] is a confidence interval for UKo with confidence

coefficient 7. Also since

"Suppose el , ••• , ~ are unknown parameters and

[ell' e12 ], ... , [ehl'~] are 100 [l-<l~C)] % confidence intervals

for el

, ••• , \ respectively. Then the probability is at least c

that these confidence intervals simultaneously contain el , ••• , \

respectively."

[1962, p. 29:0, Who also gives a proof of the result.

that if '1a. and ~ satisfy the equations,

ok(~-i)T r 1 -n-l,~n'1a.,~1+7) - s '

is a strictly increasing function of UK' we can say that

[F(~l;(O,l»,F(~;(O,l»]is a confidence interval for

F(Uv ;(0,1» = F(~.;(~ ,0" » with confidence coefficient 7 •nO K 0 0

We are going to use a result due to Tukey. The state-

I

I.IIIIIIII_IIIIIIII·I

(the true value of UB) •

Thus suppose uAJ. and uA2 satisfy the equations,

In(xA.-X)Tn-l,..fuuAJ.' (1-c)/4 = s '

.fa(xA-X>Tn- l ,..fuUA2,1-(1-C)/4 = s '

so that [uAJ.' u.A2] is a 100[l-i(l-c) ]~ confidence interval for UAO

•

Furtherxoore suppose ~ and ~ satisfy the equations,

so that [~,~] is a lOO[l-~l-c)]~ confidence interval for Uno •

Then, according to Tukey's result,

Pr(uAl ~ UAO ~ u.A2' ~ ~ UBO ~ ~) 2:: c •

That is, the region (UA,UB) IUAl ~ UA ~ uA2'~ ~ UB ~ ~J in the

(UA,UB) plane is a confidence region for (UAO,UBO ) with confidence

coefficient not less than c. Since UAO < UBO ' we my reduce this

confidence region to its intersection with that part of the (UA,UB

)

plane where UA < UB ' and the probability that the reduced confi­

dence region contains (UAO Uoo) will be the same as the probability,that the original confidence region contains (UAO,UBO). Since the

lIBpping, UA = (xA-lJ.)/rr, UB = (XB-lJ.)/a, from the region

«lJ.,a) I rr> OJ of the (lJ.,a) plane onto the region «UA,UB) IUA < UB)

of the (UA, UB

) plane is one-to-one, we can say

140

Pr«f..LO,CTO)e(f..L,CT) IUAl ~ (XA-f..L)/CT ~ uA2' ~ ~ (:x:a-f..L)/CT ~ '132P > 0]

= !R (say» •x

Thus Pr«f..L ,CT )e m) > c. Also m is connected and is the inter-o 0 x - x

section of sides of iso-F(x;(f..L,CT» surfaces. Hence M(!R) is a bandx

and Pr«f..L ,CT )e m ) = Pr(D(f..L ,CT ):: M(m » •o 0 x 0 0 x

We propose to use that part of M(mx ) in which xA ~ x ~ :x:a ' as

a confidence band for that part of D( f..Lo' CTo) in which xA ~ x ~ :x:a •

It is convenient to have some notation designed for this idea.

Therefore if [Jl].,~] is an interval on the x-axis and S is a set

of points in the (x,y) plane, we will denoteSl (x,Y)lxe [Jl].,~])

by S I[Jl].,~]. We can now say we propose that M(!\)/ [xA,:x:a] be used

as a confidence band for D(f..Lo' CTo ) / [xA,:x:a]. This proposal seem

reasonable in view of the following considerations. First, we note

that M(mx ) I[xA,:x:a] is a band since M(mx ) is a band. Next, we note

that m can be expressed asx

mx = ((f..L,CT) IF(uAl;(O,l» ~ F(XA;(f..L,CT» ~ F(u.A2;(O,l»,

F(~;(0,1» ~ F(XJ3; (f..L, CT» ~ F('132; (0,1») ,

which means that M(m ) consists of the graphs of all normal c. d. f. r sX

with ordinates at ~ and XJ3 that fall in certain intervals expressly

formed to serve as confidence intervals for the true ordinates at

XA and :x:s. Besides tending to narrow the band at xA and XJ3 IOOre

than elsewhere, these requirements would also intuitively seem to

have a constricting effect on the band, particularly for those

points between xA and XJ3; that is, for xe[xA,XJ3]. As for the

II

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_IIIIIIII

••I

I

I.IIIIIIII_IIIIIIII·I

141

confidence coefficient of M(!RX

) I[xA'~]' it is the same as tmt of

M(!R). This is eq,uivaJ.ent to saying thatx

Pr(D(~o,ero) C M(!Rx » = Pr(D(~o,ero)1 [xA'~] C M(!RX) I[xA'~]) •

We will prove the above statement, by showing that

D(~o,ero) C M(!Rx) ~ D(~o,ero)l [xA'~] C M(mx) , [xA'~] •

It is clear that D(~o,ero) C M(!Rx ) -+ D(~o,ero)1 [xA'~] C M(!Rx) I[xA'~].The converse follows from the nature of !R • Explicitly,x

D(~O' ero) j[xA'~] C M(!Rx) I[xA'~] -+

F(uAl;(O,l» ~ F(xA;(~o,ero» ~ F(uA2 ;(O,l»,

F(~l;(O,l» ~ F(xn;(~o,ero» ~ F(~;(O,l»

Let us go into the details of constructing M(!Rx) for the

situation where 0 < uAl <~ < uA2 < ~2. We have that

!Rx = {(~,er) l(xA-~)/uA2 ~ er ~ (xA-~)/UA1' (~-~)/~ ~ er ~ (~-~)/~l)

(see Figure 4.1).

As in previoos band constructions, it suffices to consider the

boundary of !R.x, <B.eRx). Some analytic geometry will convince one

that CB(m.x:) is the union of the four sets, CBa, ~, (Bc' and CBd, where

CBa = {(~, er) I er = (xA-~)/uA2' er ~ (~-xA)/('132-uA2) = era (say)) ,

~ = {(~, er) I er = (~-~)/~, era ~ er ~ (~-xA)/('132-uAl) = Cb (say)),

CBc = {(~, er) I er = (xA-~)/uA1' erb ~ er ~ (~-xA)/(~-UA1) = erc (say»),

CBd = {(~, er) I er = (~-~)/~l ' erc .5 er ).

142

Figure 4.1 C.d.f-wise exhaustive confidence region on parameters(Ilo' 0'0)_of a normal distribution for values in thein-cervaJ. [:K.A'~].

On <Ba

, (x -Il)/O' = (x-xA)/er + uA2 ' which is a non-decreasing

function of er if x < xA' and a non-increasing function of er if

xA

S X. We note that lim [(x-xA)/er + uA2 ] = uA2 • Henceer -'00

inf«x-Il)/er) occurs at (xA-uA2O'a = Ila (say ), era);<Ba

ifxASX, sup«x-Il)/O') occurs at (Ila ,era ),<Ba

inf«~-Il)/0') = uA2 •<Ba

II

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_IIIIIIII

·1I

143

lim [(X-~)/O' + ~l] = ~l. Hence0'-+00

Since one end point of 11t, (~, 0' ), is also an end point of() a a

If x < xA ' sup ( (x-~)/ 0') occurs at (~c'O'c);lBc

if xA .:s x , inf( (x-~)/0') occurs at (~c'O'c)·lBc

~ < x.

if x .:s ~ ,

On ~ , (x-~)/O' = (x-~)/O' +~ , Which is a non-decreasing

function of 0' if x .:s xB' and a non-increasing f'unctioo of 0' if

On lBc' (x-~)/O' = (x-xA)/O' + uAJ.' which is a non-decreasing

function of 0' if x < xA' and a non-increasing function of 0' if

xA .:s x. One end -point of lBc' (~,O'b) ,haS been considered as part

of ~. Therefore we need only deal with the end -point

(xA-unO' = IJ. (say), 0').n c c C

lBa ' and thus has aJ.ready been considered, we need only take cogni­

zance of the situation at the remaining end point (~-'132O'b = ~(say),

Ob). We note that

On lBd, (x-~)/O' =(x-~)/O' + ~l ' which is a non-decreasing

function of 0' if x < xA' and a non-increasing function of 0' if

xA-.:s x. Since the end point, (~c,O'c)' of lBd has been treated as

part oflBc ' we will only consider the situation on lBd as 0' -+ 00 •

We observe that

I

I.IIIIIIII_IIIIIIII·I

144

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.11IIIIIIIIII

_IIIIIIII

·1I

(X-~)/O'b ... ~ = (X-XA)/O'b + uAl < (X-XA)/O'a + uA2 •

Hence if x < xA' inf«x-IJ.)/O') occurs at (~'Ob) •

Suppose xA :s x :s~. Then sup«x-IJ.)/O') = max(sup«X-IJ.)/O'),'1J~(mx) ~a

= max{(X-lJ.a )/ era ' \lBl)

= max«X-XA)/O'a + uA2'~l)

= (x-IJ. )/0' •a ~

:. sup«x-IJ.)/O') occurs at (Il ,0' ) •m(m

x) a a

inf«x-IJ.)/O') = min(inf«x -IJ.)/O'), inf«x-IJ.)/O'»~(mx) ~a ~

= min«x-lJ.a)/O'a , (X-~)/Ob)

= min«x-XA)/O'a + uA2' (X-~)/O'b + ~) •

= uA2 ;

SUp«X-IJ.)/O') = max(sup«x-IJ.)/O'), Sup «x-IJ.)/O'), sup«x-IJ.)/O'»~~X) ~a ~c ~d

= max(uA2.' (x-IJ. )/0' , U )C c Bl

If x < xA'

if ~ < x,

. Since 0 < ~ <; , and uAl < uA2 ' we conclude that for x < XA 'a b

I

I.IIIIIIII_IIIIIIII·I

inf«x-IJ,)/a) = min{in:f«x-IJ,)/a), in:f({x-IJ,)/a), in:f«x-IJ,)/a»<B (m X) <Ba ~ <Be

= min(uA2, (x-~)/ab' (x-lJ,c)/ac)

= min(u.A2' (x-~)/ab + ~, (x-xA)/ac + uAl) •

Now (x-~)/ab + '132 = (x-xA)/(J'b + (xA-~)/TJ'b + '132

= (x-xA)/(J'b~2: (x-xA)/(J'c+UAJ.' since1.. > L .c;, (J'c

AJ.so (x-xA)/ac + uAJ. = (x-~)/ac + (~-xA)/ac + uAl

= (x-~)/ac + '131 < u.A2' since '131 < uA2 •

Suppose xn < x. Then

sup ({x-IJ,)/a) = ma.x(sup ({x-IJ,)/(J'),CB (mx) <Ba

= max«x-lJ,a)/aa' (x-~)/<1,)

= ma.x«x-xA)/ 0"a + uA2J (x-~) / Ob + '132) •

Now (x-xA)/aa .. 1 J·'2 = (x-~)/(J'a + (~-xA)/aa + u.A2

1 1= (x-~)/aa+'132«x-~)/ab~' since O"a < Ob •

:. sup «x-IJ,)/a) occurs at (~,ab) •<B(mx)

in:f«x-IJ,)/a) = min{in:f({x-IJ.) / 0")" in:f«x-IJ,)/a),. in:f«X-IJ,)/an<B(m) <Ba <Be <Bd

= min(u.A2' (x-lJ,c)/ac' '131)

= min(u.A2' (x-xA)/(J'c+ uAJ." '131)·

Now (x-xA)/ac + uAl = (x-~)/ac+ (~-xA)/ac + uAJ.

= (x-~)/ac + '131 > '13J. •

146

Also uA2 > '131 under our assumptions.

Hence, if JtB < x, inf'(x-~)/O") = '1n ·(B(~)

Fi.na.lly, we calclude that

M(!Rx) = {(x,y) IF(x; (~, O"b» =s y =s F(~; (0,1», x < xA} U

{(x,y) l(x,Y)€[D(~a'O"a),D(~c'O"c)J,~ =s x =s ~ }U

((x,y) IF(um.;(O,l» =s y =s F(x;(I\'O"b»' ~ < x} •

We see that in contrast to the band of Sub-section 2.4.3, the width

of which became smaller> as x tended to 00 or -00 , the width of M(!Jt ). x

increases as x goes from ~ to 00 and from xA to -00. However, M(!Rx)

is not meant to be a satisfactory band for all values of x.

Actually, in order to show that M(!Rx) I[xA,~J is a confidence

band for the set of functions F(X;(~o'O"o»' X€[xA'~)' with confi­

dence coefficient equal to that of !Jt ,we do not have to considerx

M(!Rx) in i ts totality at all. We need simply note the.t !Jtx is

exhaustive w.r.t. the functions of (~,O"),F(xA;(~'O"» and

F(~;(~,O"». That is, by Theorem 1.7, every point in the (~,O")

plane, 0" > 0, not in!Rx' has at least one iso-F(xA; (~, 0"» surface or

one iso-F(~; (~, 0"» surface through it, no point of which is in !Jtx•

.AJ.so by Theorem 1. 7, this means ~ is exhaustive w.r. t. any set ofx

functions of (~, 0") that includes F(xA; (~, 0"» and F(~; (~, 0"». Thus

the matting based on !Rx w. r. t. to any such set of functions has the

same probability of containing the true values of all the functions in

the set as!Jt has of containing the true value of (~, 0"). And if allx

the functions in tl1e set are continuous in (~, 0"), we can then say that

I

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_IIIIIIII

••I

I

I.IIIIIIII_IIIIIIII·I

the matting based on!R w.r. t. this set of functions is a band byx

Corollary l.ll. One such set of continuous functions of (~er) con-

sists of F(x; (\-l, er», x €[xA'~]. The I18tting based on!R w.r.t.. x

this set of functions is precisely M(!Rx) I[xA'~].Finally, we should note that we only have a lower bound, c, on

the confidence coefficient of M(!Rx) I[xA'XB]. This unsatisfactory

situation springs from the fact that the confidence coefficient of

!Rx itself is not precisely known, but merely bounded from below at c.

4.l.3 A Confidence Band for Percentiles of a Normal Population in

a Selected Interval

Suppose we want to form a set of simuJ.'aneous confidence

intervals on the true percentiles of a nornnl. popu..la.tion with c. d. f.

y = F(x; (\-lo' ero»( (Ilo' ero)unknown), for all values of y in the interval

[yA' YB]' That is, vIe want a confidence band on the set of :functions

of (Il,er), L = \-l+zer , Z€[zA'~]' where

Now, in order to form a confidence interval on any one percentile

\-l + zler (zl a fixed number) of a normal distribution « \-l , er )o 0 0 0

unknown), having taken a sample of size n from this distribution, we

can use the fact that ~(\-l +zler -X}/s is distributed as T l r. 0 0 n- ,~nzl .

and thus has a parameter-free distribution [Johnson and Welch, 1939].

In this sub-section, we will use a set of simultaneous confidence

intervals on the two percentiles, Il + zAer, \-l +z..er, of a nornnl.o 0 0 ~ 0

distribution to develop a confidence band on percentiles fitting the

description given above. We will proceed in much the same manner as

148

in the previous sub-section.

Thus we define the one-to-one corre spondence,

LA = J..l. + zA0" , ~= J..l. + ~O" ,

or inversely, IJ.=(~LA - zA~)

0"=(~-LA)

(~ - zA),

(z..e - zA),

between the (J..l.,0") plane, 0" > 0 , and the (LA,I.:B) plane, LA <~. We

will shortly indicate how a set of simultaneous confidence intervals

for

Iro = J..l.o + z..eJ..l.o '

the true values of LA and ~, may be found. For the time being, let

us suppose we have found such confidence intervals. That is, assume

we know

*=m (say» = c ,y

where .tAl' .tA2.' .t.m., .t.B2, are values computed on the basis of a

*sample of size n from a N( J..l. ,02) population. We can map mintoo 0 y

the (J..l.,0") plane, 0" > 0 , under the one-to-one correspondence given

at the beginning of this paragraph, to obtain a confidence region

m (say), for (IJ. , 0" ) with confidence coefficient c. Explicitly,Y 0 0

By Theorem 1.13, m is exhaustive w.r.t. the set of functions ofy .

(J..l.,O"), J..l. + z..e0" , and tms exhaustive w.r.t. any set of functions of

(J..l.,O") that contains J..l. + ZA0", J..l. + z..e0" , one such set being the set

of functions of (J..l.,O"), J..l. + ZO" , Z€(zA'~]. Let W(my ) be the

matting based on m w.r.t. to the set of f'uncticns,y

I

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_IIIIIIII

·1I

Rewriting the expressions for mA and~, we have

(mA'~)' g(mA'~) say, is parameter-free. Employing the standard

technique via the Jacobian for obtaining the probability density

..fu(1l + zAO' - i)/s ,o 0

..fu(1l + Z-.O' -' i)/s •o B 0

, otherwise,

•

,m = [~n(x-IJ )/0' ~ZA]/(s/O' )A 000

m... = [~n(x-Il )/0' ~ZB]/(s/O' )BOO 0

In forming a set of simultaneous ccnfidence intervals on LAO

and ~O' we can make use of the joint distribution of (mA, ~) ,

where

Il + ZO', Z €[zA'~]. Since all functions in this set are continu­

ous in (1l,0') and since my is connected, M' (my) is a band, and since

m is exhaustive 'tv.r. t. this set of functions, M' (m ) has probabilityy y .

c of containing the true values of all the functions in this set.

of functions of random variables, we find that

(~-ZA)n -n[(zA~-~mA)2+(n-r)(zB-ZA)2]g(mA'~) = C n+l exp( )2 ) ), mA< ~ ,

(~-mA) 2(~-mA

where

We see that mA and ~ are functions of two independent random

variables each with a parameter-free distribution, namely

,[n(x-IJ )/0' , which is a N(O,I) r.v., and s/O' , which is a .J:;;t l/(n-l)o 0 0 '11-

r.v. It follows that the joint probability density function of

I

I.IIIIIIII_IIIIIIII·

I

Distribution Functions

confidence bands on multivariate c.d.f's. However the actual con-

a confidence band on the joint c. d. f. from confidence bands on the

sider a few relatively simple multivariate situations.

I

••IIIIIII

_IIIIIIII

••I

Then a confidence band, with confidence coefficient atk

to n c. , can be formed on tile joint c. d. f. of. 1 ~~=

The general theory of Section 1 can be used to put parametric

4.2 Confidence Bands on Multivariate Cumulative

Now if m~ = (mA'~) ImAl :s mA :s mA2, ~1:S ~ :s llJ32) is such that

~, g(mA,~)dmAd.m:B = c ,y

150

struction of bands in specific multivariate cases appears to be more

Thus suppose the r. v. 's, the joint c. d.f. of Which we want to

Pr«LAO'~O)€(LA'~)Ix +uus/.[n:s LA'S x+mA2.s/.fn ,

X+~ls/.fn :s ~ :s x+~s/...frt, LA < ~» = c •

difficult than in univariate cases. In this section, we will con-

know, are tndependent. Then, under certain conditions, we can form

marginal c.d.f. 's. Precisely, let there be k independent r.v. 's

dependent.

least equal

X., i = 1, ••• , k, each with a c.d.f. F.(x.), i = 1, ••• , k. For~ ~ ~

each i, i = 1, ••• , k, let there be a confidence band Bi on Fi(xi ),

such that the k confidence bands B., i • 1, ••• , k, are also in-~

then we can say

Thus we have shown how a set of sinmltaneous confidence intervals

on LAO and ~O may be formed.

method other than that just proposed. Suppose we are interested in a

151

First, let us see row an iso-F (Xl' X2; (~, ~, 0"» surface

C«Xl ,X2)'Y) may be described. If we define

~ = (Xl-~)/~ , '2 = (~-~)/~ ,

variate c.d.f. for whiCh the r.v. 's involved are independent, by a

12;(

(Xl ,X2, ••• , Xk), F(Xl'~' ••• , ~) say, as follows. Express eaCh

B. as the collection of simultaneous confidence intervals~

[F.L(x.),F.U(X.)]' where x. assumes every real value. Then, since~ ~ ~ ~ ~ k

Xl' ••• , Xk are independent, F(Xl , ••• , ~) = nFi(xi ), and we cank k i=l

say Pr( n F.L(X.) < F(Xl , ••• , x) < n F.TT(X.), all real values.l~ J. - .K -.lJ.u J.~= k J.=

of (xl' ... , x» ~ n c...K • 1 ~J.=

Now we will try to construct a confidence band on a multi-

is a straight line section, as it is the intersection of the two

then C«~,X2)'Y) consists of all values of (~,~,~), 0 < ~ , such

that

population with a circuJ.a.r normal c. d. f. ,

(Xl-~O)/~O (~-~O)/~O

F(Xl'X2;(~o~ ~O' ~O» = 2; J-o:> ex;p(-t2/2)dt !ooex;p(-t

2/2) dt

(-00 < ~ < 00, -00 < X2< 00),

Where ~O' ~O' ~O are the true values of the parameters ~,~, (f ,

respectively.

Given anyone pair of numbers \,~, satisfying the above equation,

the set

t

'.IIII·IIII_IIIIIII••I

Its equation can be written as

152

II

.1 1

IIII,I

III-,IIIIIII.,I

(~-~)/'1 = (~-~)/~ ,

~ = (~-(V'1)xl) + (V'1)~ ·or

of cr, pass through the point (~ = Xl' ~ = ~, cr = 0). We see

that every point in c«xl'~)'Y) is on one of these straight lines.

It is also true that every point in C( (Xl'~)'Y) is on only ale of

these lines, for given a value of (~,~,cr), \ and \ are uniqu~

defined.

where (~,~, cr) is any point on s,. Thus

Spherical coordinates provide a means of visualizing c«~,~),y)

to some extent. Let the origin of such a coordinate system be at

tan 11 = V\ ·Next let 41 be the angle L makes with the (~,~) plane.

Thus

the (~,~) plane. :,' is fully specified by the tangent of 11 ,

where 11 is the angle :" makes with the ~-axis, and by the sign of

\ which indicates ~ether:" is defined for values of ~ greater

than or less than ~. We see that (~,~) in S, , satisfies

the point (~,=~, ~ = ~, cr == 0). Consider anyone of the line

sections, :, say, just discussed. Let :,' be the projection of:' in

All such straight line sections, if extended to non-positive values

We now attend to constructing one possible confidence band

say

are mutually independent. On the basis of these 3 statistics we can

153

sample values of Xl' and ~l' ••• , ~ represent sample

tancll = (f/,.)~(f2 +~a2

= 1/,.)1 + ~ •

s = J[ £ (Xl,-Xl ) + ~ (~,-x...)]/2(n-l)p i=l ~ i=l ~~ ~

It is now clear that S. can be described by the sign of '\, and the

values of ~./~ and 1/,.)10 + ~. Finally, since

1 lA:I. J~2;t exp(-t2 /2)dt. exp(-t2/2}dt = y""<lO -00

"ruled surface" [Salmon, 1874].

pr(xl-~0"0 =s ~O =s Xl + d10"0) = c1 (~= ei(l+Cl)/J"n);

Pr(~-d2crO S ~O S X2:. ~'b) = c2 (~= ei(1+C2/~);

Pr(O"l =s 0"0 =s 0"2) = c3

implicitly defines ~ as a function of '\ ' we conclude that the line

sections t. in C( (Xl'~) , y) can be indexed by values of '\. As '\

varies, the corresponding line section S. generates c«xl'~)'Y)'

That is, C«Xl'~)'Y) is a surface generated by the motion of a

straight line section and is thus by definition a portion of a

for F(xl'~;(~'~'cr». Suppose we take a random sample of size n

from the population under scrutiny here. Let "':il' ••• , ~ rep-

values of~. We recall that the statistics

n nx.. = 1: x.. ,/n, x... = E x2 ,/n ,

L '1 L~ ~, 1 ~~= ~=

resent

IIeIIIIIIII_IIIIIII·.-I

we now describe.

clear that m is connected. Hence, since F(~,~;(~,~,O"» is contin­

uous in (~,~,O"), M(m) is a band on F(Xl'~;(~O'~O'O"O»' and we

can say the confidence coefficient of M(m) is at least as great as

I

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I_I

IIIIII,I

••I

154

•2~-1)S~

£where ,~(n-l),Ml-C;)

From the above statements it follows that

0"1 :s 0" :s 0"2)

is a confidence region for (~O' ~O'O"O) with confidence coefficient

cl c2c;. mis shown in Figure 4.2 •

Unfortunately, it is not clear whether or not m is an inter­

section of iSO-F(~,.~;(~,~,O"»surfacesand hence ,exhaustive with

respect to the c.d.f. family being investigated here. Howeve:r:, it is

~l = (~,~.0")IX1-~0"~ ~§1+<\0"1;~-~0"2:S ~ ~~0"2; 0= 0"2 ) •

~2 = ((~,~, 0") I~ = ~-~0"; ~-~O":S ~ :s ~~O"; 0"1 :s 0" :s 0"2)·

~; = ((~,~, 0") I~-~0" :s ~ :s Xl+dl 0" ; ~ = ~~O"; 0"1 :s 0" :s 0"2}~

~4 = (~,~, 0") I~~+0.10';~-~0"~~5~+~+ ~-~0";0"1 :s 0" :s 0"2).

cl c2c;.

In constructing M(m) here, we can proceed as in the univariate

cases previously considered. That is, since F(~,~;(~,~,O"» is

nx>notonic in ~( or in ~) for fixed values of the remaining parameters

(~, 0") (or (~, 0"», extreme values of F(Xl'~;(~,~, 0"» as a function

of (~,~,O") JJD.lst occur on m(m), the boundary of!R, by Theorem 1.14.

m(m) is the union of the six faces of m, ~., i = 1,2, ••• ,6, whichJ.

I

I.IIIIIIII_IIIIIII,.II

C1'

IJ.].

'Figure 4.2 Confidence region 00. parameters (1J..I.0' ~O' C1'0) on 8.circular normal distribution

:J.55

156

35 = {(~,~, 0") I~-~0" :s ~ :s Xffodl 0"; ~ = :l!2-~O"; 0"1:S O":S 0"2 }.

36 = ('"1.,~,0") 1~-dlO"l:S ~:s ~~O"l; ~-~O"l :s ~ :s ~~0"~;0'=0"1}·

We will attempt to locate where max 'F(~':l!2;(~'~'O"» =m

max F (say) occurs,by considering the faces of!)t, one by one.!)t __

max F occurs at (~-~0"2' ~-~0"2' 0"2) =2e(say), since31 'F(~,~;(~,~,O"» is decreasing in both ~ and ~.

Similarly, max F occurs at c;.-~0"1' ~-~O"l' 0"1) ... ~l (say).36

32 is a union of straight line segments on each of which '"1.

and 0" remain constant and only ~ varies. For each such straight

line segment, max F occurs where ~ is smaD.est, which would be at

a point on the edge .Ql,2e of m, joining the points ~l and ~ •

3; is a union of straight line segments on each of which only

~ varies. F(~,~;('"1.,~, 0"» attains a maximum on each of these

straight line seA;ll1ents on a edge of m, which is part of 32 ' already

considered above. Hence for every value of F(~,~;(~,~,0"» on

33' there is a vaJ.ue of F(~,~(~,~,0"» on ~l'~ at least as

great.

34 is a union of straight line segments on each of which only

~ varies. F(~':l!2;(~,~,a') attains a maxinD.11l1 on each of these

straight line segments on an edge of m, which is in 35

•

But 35

is a union of straight line segments on each of which

only ~ varies, and thus F(~':l!2'(~,~,a') attains a maxinnun on

each of these segments on ~l'2e. Hence for any values of

F(~,~;{~,~,0"» in 34 or 35, there is also a value of

F(~,~;(~,~,O"» at least as great on the edge of m, ~1'2e •

II

.'IIIIIII-,IIIIIII

••I

Hence on ~1'2e'

(~-~)/CT = (~-~+dl CT)/CT = dl +(~-Xl)/CT ,

(~-~)/CT = (x2-~+d2CT)/CT = d2+(~-X)/CT

w.r.t. CT over the interval [CT1, CT2 J.

If ~ ~ ~ and ~ ~~, then F{Xl'~; ('\, ~,CT» is a decreas-

157

Thus we conclude, given any (~,~), that :max F(xl'~;('\'~'CT»m

_00-00

occurs somewhere on ~l'~ •

In a similar fashion, it can be shewn, given any (~,~), that

min F(~,~;(~,~,O'') occurs somewhere on ~'24' where!R

~ = (~-k\CTl , ~~CT1' CTl ) ,

24 = (x;.-h\CT2, ~~CT2' CT2) •

The next problem is to find exactly where on ~1'2e '

n;x F(Xl'~; (,\,o~, CT» occurs, and exactly where on ~,~,

~ F(~,~;(~,~, CT» occurs. Let us develop the problem as regards

n;x F(~,~;(,\,~,CT». A similar development would hold for

min F(~,~;(,\,~,CT».m

On ~l'~ ,

~ = ~-~CT, ~ = ~-~CT, CTl ~ CT ~ CT2 •

and we see that our problem is to maximize,

{~-~)/CT {~-~)/CT

~ J exp{-t2/2)dt J exp{-t2/2)dt

ing function of CT and ]'1'10",,: F over ~l'~ occurs where CT is smallest

in ~l'~ , namely at ~l. If ~ S xl and ~ ~ ~ , then

F{~,~ ; ('\, ~,CT» is an increasing function of CT and thus max F

over ~l'~ occurs 'Where (j' is largest on ~1'2e ' namely at ~ •

I

I.IIIIIIII_IIIIIIII·

I

Then max F over .21,§e

I.1,IIIIIIII

_IIIIIIII

••I

It is evident that additional research is needed in putting

158

confidence bands on multivariate c.d.f. t s •

and observing

Difficulties arise When xl-Xi and X2-~ have opposite signs.

In this case, it is not immediately apparent where on .QI'~ ,

max F over .Ql'~ occurs. One way of conceptualizing the problem here

is by putting

That is, we want to find the ma.xi.mu.m of a circular normal c. d. f.

we will not pursue the analysis of the situation any further.

over the (zl' z2) plane on a segment of the straight line z2 = A + Bzl •

This 'WOuld appear to require some numerical investigations. However

where

I

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5. SUGGESTIONS FOR FUTURE RESEARCH

The theory of Section 1 applies to putting simultaneous confi-

dence intervals on any set of parametric fUnctions. It may be of use

in situations other than those to Which it has been applied here.

First of all, there are other c.d.f. forms, such as the garmna and

the Weibull, to imich the method may be profitably applied. other

possibilities for future applications may be in putting confidence

bands on the differe~between two c.d.f. 's, which may be of partic­

ular interest in bioassay, and in putting a confidence band on

p. d. f. 's (rather than c. d. f. 's), as non-parametric methods for putt­

ing confidence bands on univariate p.d.f. 's already exist [Walsh,

1962].

The appraisal of SIlecific confidence bands by means of the

criteria proposed in Section 3 has only been touched on here, and

thus the superiority of the parametric bands to the K-S bands has not

been fully demonstrated. Also no attempt has been mde to find

"best" bands in any sense.

The relationship between properties of confidence regions for

parameters and the bands to which they give rise has not been explored.

For instance, for those ccnfidence regions which are a composition

of confidence intervals on a location parameter a and a scale param-o

eter bo

-- such that cl is the confidence coefficient for the confi-

dence interval on ao

and c2

is the confidence coefficient for the

confidence interval on bo--there may be a pair of numbers cl ' c2

(Where cl

c2

= c, the confidence coefficient for the confidence region

on (a ,b », that optimize whatever criterion by which we are judgingo 0

the band.

160

Much additiona! 'WOrk can be done on the topics discussed in

Section 4. More intensive inquiry is needed into the merits of the

confidence bands proposed there. It would be desirable to delineate

some non-parametric solutions to the problems posed there, and to

compare the:; e to the parametric approaches suggested. Apropos of this

is the fact that Simpson [1951] produced counter-examples showing

that a straightforvm.rd generalization of the Ko1nx:>gorov-Smirnov

confidence band to continuous multivariate c.d.f. 's is not possible.

That is, he deIOOnstrated that the maximum absolute difference between

the true distribution function and the sample distribution function

is not distribution-free in the case of continuous bivariate c.d.f. 's.

With respect to non-parametric confidence bands on univariate

c.d.f's, all possibilities have not been fully exploited. As remarked

in the introduction, the K-S band is not the only possible non­

parametric band, one could construct for a univariate c. d. f. The

ideal comparison between parametric bands and non-parametric bands

would be between the non-parametric band that optimized whatever

criterion one chooses for whatever distribution one assumes, with the

parametric band that optimizes this same criterion. If the perform­

ances of the non-parametric bands depend to a great extent on what the

form of the true population distribution happens to be, we may, by

using the non-parametric band that does "best" for the particular

distribution form that we believe the population to in fact follow,

do almost as well as in assuming this distribution form to begin with,

in order to construct a parametric confidence band.

I,

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6. LIST OF REFERENCES

ApostQJ.., T. M. 1957. Ma.theIlBtical Analysis. Addison-WesleyPublishing Company, Reading, Massachusetts.

Blackwell, D. and Girschick, M. A. 1954. Theory of Ge.nr!s' andStatistical Decisions. John Wiley and Sons, New York.

Bohrer, R. 1964. Confidence bounds for response surfaces. ResearchTriangle Institute, North Carolina.

Carlton, G. A. 1946. Estimating the parameters of a recta.ngulardistribution. Annals of MatheIlBtical Statistics 17:355-358.

Comrie, M. A. 1956. Barlow's Tables. Chemical Publishing Company,New York.

Dunnett, C. W. 1955. A multiple comparison procedure for comparingseveral. trea.tments with a control. Journal of the AmericanStatistical. Association 50:1096-1121.

Epstein, B. 1960. Estimation of the parameters of two parameterexpm ential. distributions from censored samples. Technometrics2:403-406.

Gafarian, A. V. 1963. Confidence bands in straight line regression.System Development Corporation, Santa M:>nica, California.

Hall, D. W. and Spencer, G. L. 1955. EleIOOntary Topology. JohnWiley and Sons, New York.

Johnson, N. L. 1957. A note on the mean deviation of the binomialdistribution. Biometrika 44:532-533.

Johnson, N. L. and Welch, B. L. 1939. Applications of the non­central t distribution. BioIOOtrika 31:362-389.

Kelley, J. L. 1955. General Topology. D. Van Nostrand Company,New York.

Kelley, T. L. 1948. The Kelley Statistical Tables, Revised 1948.Harvard University Press, Cambridge, Massachusetts.

Kendall, M. G. and Stuart, A. 1961. The Advanced Theory ofStatistics, Volume II. Charles Griffin and Company Limited,London.

Lyusternik, L. A. 1963. Convex Figures and Polyhedra. DoverPublications, New York•

162

Miller L. H. 1956. Table of percentage points of Ko1Joogorov statis- .tics. Journal. of the American Statistical Association 5l:111-12l.

M:x>d, A. M. 1950. Introduction to the Theory of Statistics. McGraw­HiD. Book Company, New York.

National Bureau of Standards. 1950. Tables of the Binomial Probabil­ity Distribution. Applied Mathematics Series 6, Washington.

National Bureau of Standards. 1953. Tables of Normal ProbabilityFunctions. Applied Mathematics Series 23, Washington.

Pearson, E. S. and Hartley, H. o. 1958. Biometrika Tt'.bles forStatisticians, Volume I. Cambridge University Press, Cambridge.

Pryde, J. 1961. Clta.mbersts Seven-Figure Mathematical Tables. W. &R. Chambers Ltd., London.

Resnikoff, G. J. and LiebeI"lllU1, G. J. 1957. Tables of the Non­Central t-Distribution. Stanford University Press, Stanford,California.

Roy, S. N. and Bose, R. C. 1953. Simultaneous interval estimation.Annals of Mathematical Statistics 24:513-536.

SaJ.mon, G. 1874. A Treatise on the Analytic Geometry of ThreeDimensions. Hodges, Foster and Company, Dublin.

Scarborough, J. B. 1955. Numerical Mathematical Analysis. TheJohns Hopkins Press, Baltimore.

Sheffe, H. 1959. The Analysis of Variance. John Wiley and Sons,New York.

Simpson, P. B. 1951:. Note on the estimation of a bivariate distri­bution function. Annals of Mathematical Statistics 22:476-478.

Taylor, A. E. 1955. Advanced Calculus. Ginn and Company, New York.

Thielman, H. P. 1953. Theory of Functions of Real Variables.Prentice-Hall, New York.

Wa1d, A. and Wolfowitz, J •. 1939. Confidence limits for continuousdistribution functions. Annals of Mathematical Statistics10:105-118.

Walsh, J. E. 1962. Handbook of Nonparametric Statistics. D. VanNostrand Company, New York.

WilkS, S. S. 1962. Mathematical. Statistics. John Wiley and Sons,New York.

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IIIIII,.I

163

Working, H. and Hotelling, H. 1929. Appli'cations ot: the theory ot:errors to the interpretation ot: trends. Journal ot: the AmericanStatistical Association 24:73-85.

Yaglom, I. M. and Boltyanskii, V. G. 1961. Convex Figures. Holt,Rinehart, and Winston, New York.

7. APPENDDC

7.l Theoretical Supplement to Section l

The following definitions are found on the indicated pages of

Thielman [l953].

A real number M is said to be a least upper bound ora supremum of a given set S of real numbers if

(l) M is an upper bound of S, i.e. every elelOOntof S is less than or equal to M.

(2) for every N lesstban M there exists at least ones of S such tha.t s > N. (p.l3-l4].

A real number L is said to be the greatest~bound or the infimum of a set S of real numbers if

(l) L is a lower bound of S, i.e. every element ofS is greater than or equal to L.

(2) for every number N > L there exists at least ones of S such that s < N. (p.l4]

Let a and b (b > a) be two real numbers. The set ofnumbers x such that a. < x < b will be called a closedinterval and will be denoted by [a, b]. The numbers a andbe will be called the end points of the interval. [p.l8]

The following material is found on the indicated pages of HaJJ.

and Spencer [l955].

In the study of topology, the undefined objects areusually called "point" and "open set". The relationshipbetween these objects is given by the following axioms:

Axiom 1:. Every open set is a set of point s.Axiom 2. The empty set is an open set.Axiom 3. For each point p, there exists at least

one open set· containing p.Axiom 4•. The union of any collection of open sets

is an open set.Axiom 5. The intersection of any finite collection

of open sets is an open set. [p.52]

A set S, together with a collection of subsets calledopen sets, is called a topological space if and only ifthe collection of open sets satisfies Axioms l through5. (P.53]

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165

Let 8 be a set and ~ a collection of 8. Then ~

is said to generate the collection ~ of subsets of 8defined as follows: A subset K of S is element of~ if and only if K is the union of a collection ofelemants of ~. The collection ~ is said to be a basisfor the collection ~ which it generate~ [P.53]

Let 8 be a (topological) space and p a point of 8.A subset K of S is said to be a neighborhood of p if andonly if K is an open set of 8 containing p. [p.62]

The next group of definitions will be found in Taylor [1955] on

the pages indicated.

A set 8 is called closed if its complement is open.[p.135]

If 8 is a point set, a point P is called a balndarypoint of 8 if every neighborhood of P contains at leastone point of 8 and one point of the complement C(8). Thecollection of' all boundary points of 8 is called theboundary of 8. [P. 136]

Let 8 be a point set, and let y be a point which isnot necessarily in 8. We call y an accumu..1a.tion point of8 if in each neighborhood of y there is at least one pointx which is in 8 and distinct from y. (p.481]

Two point sets 8 and 82

are called separated if thefollowing three conditions hold:

(1) Neither set is empty,(2) No point belongs to both sets,(3) Neither set has a point of accumulation belong­

ing to the other set. [p.502]

A set 8 is called connected if it cannot be dividedinto two parts which are seParated. [p. 502]

In a metric set, that is, a set every two element s of whi ch have

a distance defined between them, the OI>en sphere with center Po and

radius Po is the set of all elements at a distance less than Po

from p •o

When EP (p-dimensional Euclidean space), a matric set, is re-

garded as a topological space in this thesis, the basis for the set

166

of all open sets in it is taken to be the set of all open spheres in

EP•

7.2 Theoretical SUpplement to Section 2

7.2.1 Convex Figures

When we speak of a convex figure here we mean a plane convex

figure.

A figure is said to be convex if it entirely centainsall segments containing any two of its points. [Lyusternik,1963, p.l]

Consider the straight line kb + a = h (k, h any real numbers)

in the (a,b) plane.

Its upper side is defined as (a,b) Ikb + a ~ h}.

Its lower side is defined as ({a, b) Ikb + a 2: h}.

The straight line b = bl has an upper side defined as

({a, b) Ibl 5. b} and a lower side defined as {{a,bib 5. bl } •

Let Q be a convex figure •

We shall call the line r a support line to the figureQ if (a) all of the figure Qlies on one side of the liner and (b) the line r has a point in common with the bound­ary of Q. [Lyusternik, 1963, p.7]

A E!El. is a set consisting of a certain point and all points in

a given direction from it.

Let us consider a point P on the boundary of the convexfigure Q. From this point draw all possible rays connect­ing P with points of the figure Q, and denote by R the setof points lying on these rays. [Lyusternik, 1963, p.9]

R is the tangent angle to Q at the point P. The s ides of R are the

tangent lines to Q at P. It can be shown that the tangent lines to

Q at P are also support lines to Q at P.

Let us regard the whole (a, b) plane as a topological space. By

a closed region in the haJ.:f' plane ({a,b) lb > o} , we mean a region

I

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167

that is closed if viewed as a sub-set in the ha.1f plane {(a,b) Ib > O},

taken as a sub-space of the Whole {a, b} plane. See Hall and Spencer

[1955, p.7l-'72] for a definition of the term "sub-space". A region

may not be closed in the whole {a, b} plane, but it may notwithstanding

be closed in the topological space constituted by the half plane

{(a,b}lb > oj as a sub-space of the whole {a,b} plane. Thus, though

sides of straight lines in the half plane {(a,b) Ib > oj, that is,

sets like (a,b) Ikb + 60 S h, b· > O}, (a,b) Ikb+a 2: h, b>O J, and

(a,b) 10 < b S bl ), are not closed sets in the whole (a,b) plane,

they are closed sets in the half plane Ha, b} Ib > oj.

Theorem on the Formtion of Convex Figures

Any closed bounded convex region in the half plane ({a, b) Ib > 0)

can be formed as an intersection of sides of straight lines in the

half plane (a,b) Ib > 0 J •

~: From Yaglom and Boltyanskii [1961, p.133], we learn that

every bounded convex region in the whole (a, b) plane, that contains

its boundary, is the intersection of finitely or infinitely many sides

of straight lines. Now a set is closed if and only if it contains

its boundary [Kelley, 1955, p.46]. Hence we can say every closed

bounded convex region in the whole {a,b} plane is the intersection

of finitely or infinitely many sides of straight lines.

We next observe that every closed bounded cmvex region mH in

the half plane Ha, b} Ib > O} can be expre ssed as the intersection of

the half plane ({a,b) Ib > oj with a closed bounded convex region!)lW

in the whole (a,b) plane simply by considering !Ji:I as a sub-set of the

whole (a, b) plane and then taking 9tWto be the union of 9tH and its

~68

boundary in the whole (a, b) plane. Then since !Rw is the intersection

of sides of straight lines in the whole (a,b) plane, !JtH

must be the

intersection of sides of the se same straight lines in the ha~ plane

((a,b) Ib > 0).

We will nOvT present some slight modifications of material taken

from the indicated pages of Blackwell and Girschick [1954], on convex

figures in EP, p-dimensional Euclidean space.

A subset E of EP is called convex if for any two points ~

and !e of E, the entire line segment joining ~ and !e is con-

tained in E; i.e. A~ + (l-A)!e€ E for 0 ~ A~ 1. [p.3l]

If E is a convex set of EP, a numerical function defined on E

is said to be concave if for all !.l'!e€ E and all A, 0 ~ A ~ 1, we

have

Theorem For any concave function f defined on a closed convex

p ~~subset E of E , the set T of all points (!oJ y) € .l!.- with !. € E,

y ~ f(!) is convex; if an addition f is continuous, T is closed. [P.39]

We remark here that the intersection of any number of convex

sets is also convex. [p.42] It is particularly relevant to showing

that !JtL+ is convex in Sub-section 2.4.4, to note that if p =1 in

the above theorem, then the intersection of T with the half plane

({x,y) Iy ~ 0 ) ( a ccnvex region) is also a convex region.

Theorem Let a function g be <D ntinuous an the closed interval

[c,d]. (At c, g need be continuous from the right only, at d from

the le:rt only.) Uso let g possess the first two derivatives

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169

throughout the open interval (c,d). Then g is concave on [c,d], if

the second derivative of g is less than or equal to zero on (c,d).

[p.39]

7.2.2 Theorems Used in SUb-section 2.6.4

Mean Value Theorem for multiple integrals (adapted from Apostol~

Assume that g(2f) and f(!) are :aiemann-integrable on a bounded

set min EP and suppose that g(2f) ~ 0 for each! in m. Let

m = inf f(!.> , M = sup f(!). Then there exists a real number" inm m

the interval m S " S M such that

1 f(2f) g(2f)~ = "1 g(!.) dxm !R

In particular, if p = 2, we have

m a(m) S 1 f(!) ~ S Ma(!R)m

where a(m) is the area of !R.

Theorem Let!R be a F{x;{IJ.,w»-wise convex region in {(1J.,w) Ig(lJ.,w»O}.

Then the absolute vaLues of the slopes of the tangent lines at a

balnd.a.ry point of !R must be in the interval [2,00] •

Indication of Proof: We offer tlE following heuristic argu-

ment in support of the above theorem.

By definition,an iso-F(x; {IJ., w»-wise convex region is the inter­

section in the half plane {( IJ., w) Ig( IJ., w) > o} of sides of straight

lines with absolute slope values in the interval (2,00]. Since

has a boundary consisting of line sections with slopes of absolute

170

value 2, the boundaries of the regions being intersected to form!Jt

cons ist of line sections with absolute slope values in the interval

[2,00]. If the number of such regions being intersected is finite,

the boundary of the intersection must also consist of line sections

with absolute slope values in [2,00], and we see tlRt the theorem holds

in this case. If the number of such regions being intersected is

finite, it would seem an argument, similar to that just given but

employing limiting processes, can be concocted to prove the theorem.

Theorem 2.7 (See Sub-section 2.6.4 for a state~t of the theorem.)

Proof: S(~,~) cannot be entirely within I)(~,~), as if it

were, it could not be a c-figure. Hence there must be a pair of points

both with the same w-coordinate wf 2= <»:r(~' ~), one on each side of

the line I..l = I..lt, , where the bcundary of S(~,~) crosses the boundary

of I)(~, ~). Let us refer to the member of this pair of points on the

left boundary of S(~,~) as (I..lf,wf ). To show wf = wt ' it suffices

to Show that the left boundary of S(~,~) cannot intersect I)(~,~)

at an interior point with an w-coordinate greater than wf • Proceed­

ing to prove this by contradiction, let us suppose the left boundary

of S(~,~) does intersect ;Q(~,~) at an interior point

(I..l , W), W > wf

• It is geometrically apparent that the line segmentg g g

(l..lf,Wf ), (l..lg,Wg) must have a positive slope less than 2. Next we

note that the left boundary of 8(~,~) between W= wf and W= W~ - g

must have a support line t h (at a point (~,~) say) parallel to

(1..l"",Wf ), (1..l,W ). We observe that (Ll,W. ), (I..l,w) has a positive.L g g . on n g g

slope less than 2.

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171Thus the tangent line to 8(1-1>'~) at (~, (~) and between lh and

(~, ~), (Ilg' Wg ) also has a positive slope less than 2. But this

contradicts our assumption that 8(1-1>'~) is a c-figure and hence

possesses tangent lines at every boundary point with absolute slope

values in the interval [2,00].

Theorem 2.8 If all horizontal lines intersect a closed convex

region as line segments, which are translated in a direction parallel

to the horizontal axis until their mid-points lie on any specified

vertical axis, then the figure cOIlIj;X)sed of the translated line seg-

ments is also convex.

Proof: Let 3' be the original ccnvex region referred to in the

statement of the theorem, and let 3' be the figure composed of trans-

lated line segments belonging to 3 with mid-points on a specified

vertical axis. With every two of these line segments in 3' we may

associate a trapezoid, the upper and lower bases of which are formed

by the two line segments, and the sides of which are formed by join-

ing the left end points and right end points of these two line seg-

ments.

Now we will show 3' is convex by arriving at a contradiction

through assuming it is not convex. To say a figure is convex is

equivalent to saying that for every two horizontal line segments

belonging to the figure, all points in the associated trapezoid are

in the figure. Thus, if we suppose 3' is not convex, there must be

two horizontal line segments Ll ,~ in ;;' such that at least one

point ~ , say, in the associated trapezoid T' (Ll ,L2

) is not in 3 '.

172

Let L be the horizontal line segment in 3' with the same ordinate aso

Eo. By definition Eo is not in 3'. Let M~ be the horizontaJ. line

segment in T' (L:J., ~). consisting of all points in ,T' (Ll'~) with the

same ordinate as Eo. By definition J20 is in Mo •

Let us denote the length of a line segment L bY'r(L). Now

-r(M~) > "I:(L0). To see this, note that:;' is synunetric about a verti­

cal axis. Thus M' and L have the same mid-point, and since M' and000

L are closed line segments (:; being closed), X(M') < X.(L ) implieso ,0 - 0

M~ C Lo' which ~ans J20 is in Lo and hence in :;'. But by definition,

Eo is not in :;'.

Next we observe that no matter how we translate Ll and ~ in

a horizontal direction, we can associate a trapez.oid T(Ll'~) with

them, constructed in the same manner as T' (Ll'~). FurtherIOOre, the

horizontal line segment Mo' say, in T(~,~) consisting of all po:ints

in T(Ll'~) with the same ordinate as Eo ' will ~ve the same length,

namely X(M~), as the line segment M~ in T' (Ll'~). In proving the

last statement we denote the area of a figure 30

by a(:;o). "~So

denote the di.fference between the ordinate of Eo and that of points

in ~ by D2--the ~fference between the ordinate of J20 and that of

points in Ll by Dl • Then from elementary geometry,

However,we can obtain an expression for a(T(L:J.'~» involv:lngm (Mo)'

if we note that Mo divides T(~,~) into two trapezoids, so that

a(T(Ll'~» equals the sum of the areas of these t~ trapezoids.

Thus

I

••IIIIIII

_IIIIIIII

••I

Proof: We will continue to use the notation developed in the

so that :J and 3' have the same area. FurtherJOOre, since:} is a

Due to the manner in Which 3' is formed from ~ , we can also say

~1 X(L( w) )aw •'\

a(d) =

If we set the two expressions for a(T(Ll'~» equal to each other

and solve for X(M ), we obtain a value that is the same, regardlesso

of how Ll and ~ are translated in a horizontal dire ctioo.. Hence

X(MO

) = t(M~) ,

so that X{M ) > t(L ), where M is defined on the basis of theo 0 0

relative position of Ll and ~ in the original c~vex region:} •

But since :} is convex, MeL where L is in ~. Hence000

X.(~) :s t(Lo). This is the contradiction we have been seeking. We

can conclude ~' is convex.

173

c-figure, then the new figure, formed from it, provided it is in

{(Il,w) Ig(ll,w) > 0 ) , is also a c-figure with the same area.

Theorem 2.9 If the original convex region in Theorem 2.8 is a

follows that

bound of ordinate values of points in 3 be denoted by tAt, and let UM

be the least upper bound of ordinate values of points in 3. It

proof of Theorem 2.8. Also let us regard the figures being discussed

as being in a (11, w) plane. Then let the horizontal line segment in

3' with ordinate w be denoted by L(w) and let the greatest lower

I

I.IIIIIIII_IIIIIIII·

I

c-f!gure,and

174

n-1g(lJ.,w) = n(n-1) r n+1 ' (1J.,w)e«lJ.,w) Ig(lJ.,w) > 0) ,

W

Now if we can show that 3' is a F(x;(IJ.,w»-wise convex region,

the theorem is proved. Suppose we can show that the tangent lines

to 3' at any boundary point have absolute slope values in the

interval [2,00]. Then if we regard :I' as the intersection of the

appropriate sides of its tangent lines, it is clear from Sub-secticn

2.6.3 that any such figure can also be formed as the intersection of

sides of iso-F(x;{IJ.,w» surfaces.

Thus consider the tangent lines to 3' at a boundary point

(~, ~). The slope values of these tangent lines may be viewed as

1imi.ts arising in one of four ways now to be described. Let us

denote the horizontal line segment in 3' with ordinate Wby L'(W).

It is clear that if (~,~) is on the boundary of 3', it is either

the left or right end point of L' (UH). If' (~,~) is the left end

point of L' (~), we may define the slope of one tangent line at

(~,~) as a limiting value for the slope of a secant through

(~,~) and the left end point of a horizontal line segment in :J'

with ordinate greater than ~ , as this ordinate a.pproaches '11 •If' we consider in the same manner a horizontal line segment in 3'

I

.­IIIIIII

_IIIIIIII

••I

•

1 g{ IJ., w)dlJ. dw3

1~ n-1~ n{n-1) :n+l X(L( w» dw=

= 1 g(IJ.,W)dlJ. dw3'

c =we see that

I

I.IIIIIIII_IIIIIIII·I

175

with a smaller ordinate ~, we can define the slope of what may be

another tangent line at (~IH'~). Similarly if (~,~) is the right

end point of L' (UJi), two other definitions (substituting "right" for

"left") may be made of slopes of tangent lines at (~, '1I).

We will assume ('1f~) is a left end point, and not also a

right end point, of L'(~). Let \ denote the slope of the tangent

line at ('11,~) defined in terms of a line segment L'(wG),wG > ~ •

We will show that 1\I is in [2,00]. A similar argument would hold

if we assumed any of the other three situations. Let ~l and IlG2

be the left and right end point abscissas respectively of L(wG) in

CJ and let ~ And ~ be the left and right end point obscissas

respectively of L(~) in CJ. Then put

L\L = %1 - ~ ,

~L=%2-~ ,

Since CJ is F(x;(Il,w»-wise convex, the a.bsolute slope vallles of

the tangent lines to its boundary are in the interval [2, co]. Then

viewing the slopes of these tangent lines as limiting values of

secant slopes, we can say that

-t:s lim (~L/bJJl) :s t ,6W -40

-t:s lim (~L/6.W):S t ·6.w -40

From the symmetrical fashion in which CJ' is formed, we see that

176

Thus 'H = lim 2[(~L/6w) - (~L/6w)rl6woo+O

== 2[ lim (~L/6w) - lim (~L/6w)rl6w-+O 6w-+O

Then if we consider all the possible values that lim ('\L/6w) and6w-+O

lim (~L/6w) can assume, we arrive at the conclusion that 1\1is6w-+O .in [2,co], and the proof of the theorem is complete.

We note that the argument just employed can be extended to

show that if the absolute slope values of the tangent lines to the

original figure:J were all in the interval [v, co], where v is any

non-negative real number, then the absolute slope values of the

tangent line s to the new figure :J I formed fran :J would also all be

in the interval [v, co].

I

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I

I.IIIIIIII_IIIIIIII·I

1777.3 CO!putations for Tables

7.3.1 COm;Putations for Table 3.1

[Note. In the description of computations in this thesis,

where the context makes clear what is meant, the value of a certain

number is often spoken of, when to be precise, one should speak of

the approximate value, as found from previous calculations, of this

number.]

E(WM) for the K-S band was computed by doubling the value of

D 95 given in Miller [1956], and then rounding the result to 4 d.pn,. .(decimal places).

For E(WM) of the parametric band, when only 1-10

is unknown, the

following formula was derived in SUb-section 3.3.2..1.

t.975/~ .J .~XP(_t2/2) dt •t. 975 /.Jn

t. 975 /.fn was obtained by multiplying 1.9600 by l/..[n taken, as found

in Comrie [1956], to 7 d.p., and thenrounding the result to 4 d.p.

The value of the integral in the expression for E(WM) was then found

in National Burea.u of Standards [1953] and rounded to 4 d. p. to serve

as E(WM). (As regards the erratic value 57.99 with an asterisk over

it in Table 3.1, the calculations leading to this particular result

were rechecked without uncovering an error. Perhaps some errors in

tables used for computing were compounded here to produce the devia.tion

of this value from the trend in its column.)

For E(WM) of the parametric band, when only lTo is unknown the

following formula was derived in Sub-section 3.3.'2.2.

standard normal curve within distances A and B around the origin

results being correct to a unit in the last dec imal place carried.

Then in National Bureau of Standards [1953], the areas under the

I

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_IIIIIIII

••I

178

,.tIl r. 95r 5-1.9

.tIl r. 951-1!r. 95

exp(-t2 /2) dt

where r. 95 = ~-1, '.975/x;;,-l, .025 •

The percentiles of x;;,-l involved in the expression for r. 95 were

found in Pearson and Hartley [1958]. The result of dividing one per­

centile into the other in order to obtain r.95

was rounded to 8 d.p.

Next .en r.95

was found in two stages. First the conunon logarithm of

r.95

was found through linear interpolation in Pryde [1961]. Then

the ensuing result, rounded to 8 d.p., was multiplied by

.en 10 = 2.30258509, and the product was rounded to 8 d.p. to yield

.en r.95

• Then the results for (1-1/r.95

) and (r.95

-l) were found to

8 d.p. and divided into .tn r.95

to yield quotients which were rounded

to 8 d.p. The squa.re roots of these quotients are the upper and

lower limits of the integral constituting E(WM),. The square root act­

ing as the upper limit, A say, was taken to 3 d.p. and the square

root acting as the lower limit, B say, was taken to 4 d.p., the

were found and rounded to 5 d.p. Onehalf the absolute value of the

difference between these two results was rounded to 4 d. p. and taken

as E(WM).

For E(WM) of the parametric band, When both !-Lo and 0"0 are un­

known, the following formula was derived in Sub-section 3.3.2.3.

computations, with the exception of the final results which were

volved being rounded off to 8 d.p., before being used in further

In the computations for (Xq-~)/CT2and (Xq- 11)/0"1 now to be described,

natural logarithms were calculated as for the band previously dis-

rounded to 4 d. p. Then in National Bureau of Standards [1953], the

areas under the standard normal curve within distances (Xq-~) / 0"2

179

,

Kr. = 1J..:a

"n-l,.99

,1Ku=

";:-1,.01

-~.j (e. 9847/"')2 + «Ku-KJ!(Ku+~) )In(KfI~) ] ,

(Xq-~)/O"l = (Ku-KJ-l[~(t.9847/J"n)

- KuJ(e. 9847/1.];;)2 + ((Ku-~)/(Ku+Ki» .tn(~/~) ],

(x -1-lt')/0"2'E(WM) ="cf q exp(- t 2 ) dt

(Xq-\-Ll)/O"l

where (Xq-\-L4)/ 0"2 = (KU-~) -1[Ku(e•9847/-k)

and

cussed, and any square root was so calculated as to be correct to a

unit in its last decimal place. The percentiles of ~-l involved in

the expression for E(WM) were found in Pearson and Hartley [1958].

Their reciprocals were found to 8 d.p. to yield ,~ and ~ and the

square roots of ,these reciprocals were taken to 8d.p. to yield KLand

KU. e.9

847 = 2.16223587 was found in Kelley [1948]. To obtain

e.9

847/J"n , e. 9847 was multiplied by l/J"n as given to 7 d.p. in Pryde

[1961], and the ensuing result was rounded to 8 d.p. Then the

calculation of (Xq-~)/0"2and (Xq-I\)!sl proceeded according to the

formulas just given, with the result of each arithmetic operation in-

I

I.IIIIIIII_IIIIIIII·I

180

and (Xq-~)/O"laround the origin were found and rounded to 5 d.p. For

n equal to 10. 15, 20, 24, the average of these two results rounded

to 4 d.p. serves as E(WM). For n greater than 24, one half the

absolute value of the difference between these two results was rounded

to 4 d.p. to serve as E(WM).

7.3.2 Computations for Table 3.2

For discussion of the calculation of E(W(zo;Bx_s»' let

p = F(z ), so that z =P_ Also let l-p = q •o 0 0 ~o 0 ~

Computations for E(W(zo;BlC_S» were performed before the author

knew of Gruder' s formula, so that the computations were done using the

following simple formula readily apparent from the original expression

for E(W(zo;BK

_S» given in SUb-section 3.4.

[

nl

E(W(zo;BlC_S» = Dn,.95 k:O (~) p~ <lon-k}

n2 n+ E (~)p~ ~n-k) + E (~)p~ ~-k) ]k=nl+l n2+l

nl n+ E ! (n

k) pok qno-k } + E ~«n)pk n-k),

k=O n k=~+1 n k 0 ~

where n = 24, nl = 6, n2 = 17.

D 95 = .26931 comes from Miller [1956]. The individual. termsn,.(~)p~ %-k) to 7 d.p!', as found in llational Bureau of Standards

[1950], were added together to form the three sums, each taken to

k7 d.p., within the brackets preceded by D 95. The terms - andn,. n!!.::!. were found to 7 d.p. other computations proceeded as indicated

n

in the formula just given with the result of each arithmetic opera-

tion involved being rounded to 7 d.p. before being used in further

I

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_IIIIIIII

••I

t. We find that

181

calculations. The final result of the formula was rounded to 2 d.p.

of T'p,o

.10, .04,

•

,

,

,e-Ml +C

l)

:fne-Ml +c

l)

~

T+n-l,~n Zo

T+n-l,~n Z ­

o

E(W(z ;R_» = [l-~(~n Z )]E[~(U-)-~(L-)]o -N 0

+ ~(~n Z ) E[~(U+)-~(L+)] ,~..-- 0

U-=I~-1,Ml-C2) T- r + eMl+Cl)( 1) n-l,,,,n Zo r"", n n- ",n

~-1,~1+c2) e~l+cl)T- r -

n(n-l) n-l,,,,n Zo ~I .

+_J~-1,~1+C2).U - n(n-l)

~-1,~1-c2)

n(n-l)

where

by g •o

and .01.

to serve as the value for E(W(Zo;~_S» shown in Table 3.2.

For discussion of the calculation of E(W(Zo;Bm) let ~ be the

c.d.f. of e , a N(O,l) random variable.

Now the following form for E(W(Zo;~» was derived, though not

written out, in Sub-section 3.4.

For Table 3.2, E(W(z ;~» was evaluated only for n = 24.

Resnikoff and Lieberrnan[1957] give values of g,the p.d.f. of Tp/.J23,

where T is a non-central t-random variable with non-centralityp

.J2fj. e and 23 degrees of freedom. In order to use the Resnikoff andp

Lieberman Tables, let us put Z = e and denote the p. d. f.. 0 Po '

Table 3.2 gives E(W(z ;R_» for p = .5, .25, .15,o -N 0

Also let us regard U- and L- as functions of T~3 J24 z /J23,. '0

+ + . + I~and U and L as funct~ons of T23,.J24 Z 1"'23 • Denote the dunrny_ 0 + ~

variable standing for values of T23,.J24 Z /../23 or T23,.J24 Z /"'23 byo 0

I

I.IIIIIIII_IIIIIIII·I

The integrals in the above expression were evaluated by means

Then both these quantities were taken as zero for purposes of compu-

I

••IIIIIII

_IIIIIIII

••I

E(W(z ;il_» = 109 (t)[~(U-)-~(L-)]dto~ -eo 0

+ I COg (t)[~(U+)-~(L+)]dt •o 0

LCO

g (t)dt = ~(J24 z ) ,000

we see that

Then since r 0g (t)dt =1-~(J24 z ) ,£CO 0 0

tation. This information concerning to and to+mh was deduced upon

of Simpson's Rule [Scarborough,1955, p.132-133]. Simpson's Ruleto+mh

gives a way to approx:i.ma.te It y dt, m an even integer, 0 < h,o

where y is a function of t. In our application of the rule,

go(t)[~(U-)-~(L-)]or go(t)[~(U+)-~(L+)]plays the role of y.

For P =. 5, the non-central t-random variable involved ino .

the expression for E(W(Zo;!ii)} becomes Student's t. Since the p.d.f.

of Student's t could mt be found tabled anywhere and therefore had to

be computed by the author, the calculation of' E(W(z ;il_)} f'or p =.5o -llJ 0

differs somewhat from the calculations for other p and will be. 0

treated separately.

182

First we will deal with Po other than .5. In evaluating

rCO + +J_ g (t)[~(u )-~(L )]dt for such p , t and t +mh were selected soo 0 tOO 0

that fOg (t)[~(U+)-~(L+)]dt < .00015 ando 0

ItOO

+mh go (t) [~(U+)-~(L+)]dt < .00015.o

I

I.IIIIIIII­IIIIIIII·I

183to

observing that J g (t)dt < .00015 ando o.

and J; +mh go(t)dt < .00015, according to the Resnikoff and Lieberman0++

Tables. Since 0 :s [w(U )-~(L )] :s 1 for all 'V8J.ues of t, the deduc-0-_

tion follows. J g (t)[~(U )-~(L )]dt was only calculated when- co 0o

[;, go(t)dt > .00005 according to the Resnikoff and Lieberman Tables.

This only occurred for Po = .25. In all other cases

JOg (t)[~(U-)-~(L-)]dt was taken as zero. For p = .25, the- co 0 0

Resnikoff and Lieberman Tables indicated

-.20 -.20J g (t)dt < .00005. So J g (t) [~(U-)-~(L-)]dt was regarded~ 0 ~ 0

as zero, and JOg (t)[~(U-)-~(L-)]dt, after being e'V8J.uated by-.20 0

Simpson's Rule (t = -.20, t +mh = 0), was taken as an approximationo 0

to JOg (t)[~(U-)-~(L-)]dt.-co 0

The pertinent data on how Simpson's Rule was applied are given

below.

Evaluation of Evaluation of

£co go(t)[~(U+)-~(L+)]dt JOg (t) [~(U-)-~(L-)]dt;,,00 0

Wo t m h t m h0 0

.25 0 22 .10 -.20 4 .05

.15 .10 28 .10

.10 .30 30 .10

.04 .70 34 .10

.01 1.20 40 .10

Now the com;putation of the y - values used in Simpson r s Rule :is

described. To save effort, instead of getting c equal to .95 as

should have been done to obtain strict comparability with

E(W(zo;BK_S»' c was set equal to .950012, as in this case, suitable

184

values of ei(Cl+l)' ~-1,i<1-C2)' ~-1,i<1+C2) could be directly

obtained from available tables. Thus the confidence coefficients

used were

cl = .9694 ,c2 =·98 ,

c = cl c2 ... •950012 •

This selection of confidence coefficients also held for Po ... .5 •

Table 7. 1 outlines further steps in the procedure. In writing

the table, a group of similar steps has been indicated by the symbol

si' where s is a specific positive integer. The range of i, which

is also a positive integer and the index for steps within the group

s ., is given in the step description. Lat'er references to such a~

group of steps, unless otherwise qualified, apply to all members of

the group. The result of a certain step s or si is denoted by

R(s) or R(si ) , respective ly. The column labelled "No. of d. p. " gives

the number of decimal places retained in the step result. "C." in

the "Source" column means the srithmetic operations indicated in the

corresponding step description were performed by the author with the

aid of an office calculator. Nat '1. Bur. of St. is an abbreviation

for Jlational Bureau of Standards.

Values of [~(U+)-~(L+)] and [~(U-)-~(L-)] were only computed

for t equal to same integral multiple of .05, since the Resnikoff

and Lieberman Tables give go(t) only for such t. Two sets of tables

were used to evaluate ~. When the argument, v say, of ~ was less

than 1 in absolute value, the Bureau of Standards' Table was used as

this gives [~(v) - ~(-v)] accurate to 15 d.p. for v equal to any

integral multiple of .0001 in the interval [0,1]. However for 1 < v,

I

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I

I.185

Table 7.1 Procedure for computing ordinates in appJ.1cation of

ISimpson's Rule to evaluation OfE{W{Zo;l\i» forPo = •25, .15, •10, .04, and •OL

Step Step description 1'0. Source

I d. •

8 Kelley [1948]1 t. 9847

I 2 ~3, .994 Pearson and Hartley[1958]

I3 ~3,.01 5 Pearson and Hartley[1958]

4 l/~n 7 Comrie [1956]

I 5 R(1)R(4) 8 C.

6 .JR(2) 7 C.

I 7 JR(3) 7 C.

8 R(6)R{4) 6 C.

I 9 R(7)R(4) 6 C.

1_ 10i .10 i R(8)+R{5); 0 ~ 1 ~ 4 4 C.

11i .10 i R(8)+R{5); 5 ~ i ~ 52 5 C.

I ]2. l.10i R(9)~{5) I; 0 < i < 22 4 C.J.

13i .10 i R(9)-R(5); 23 < i < 52 5 C.

I- -

14i ~(R{10i»-~{-R(10i» 8 Nat 1l.Bur.of st. [1953]

I15i ~(R(lli» 4 Kelley [1948]

16i ~(R(12.»- ~(-R{12.» 8 Nat t l.Bur. of St. [1953]J. J.

I 17i ~(R(13i» 4 KelJ.ey [1948]

18i .5[R{14.)+R{16.)];- 0 < 1 < 4 8 C.

IJ. J. --

19. R(151)+.5R{16i )-.5; 5 ~ i ~ 6 9 C.J.

I201 R{15i)-.5R{16i)-.5; 7 ~ 1 ~ 22 9 C.

21. R(15.)-R(17.); 23 < i < 52 4 C.J. J. J. --

I 22i R(181)go(·10i ) 8 C.

I·23. R(19.)g (.10.) 8 C.

J. J. 0 J.

I

186

Table 7.1 (continued)

Step Step description .0.0 Sourced.

24. R(20i )go(.10i) 8 c.J.

25 i R(21i )go(·10i) 8 c.

26. .05i R(9)+R(5); -4 ~ i ~ 0 4 c.J.

27i 1.05i R(8)-R(5) \; -4 ~ i ~ 0 4 c.

28i ~(R(26i»-~(-R(26i» 8 Nat'l. Bur. of St. [1953]

29i ~(R(27i»-~(-R(27i» 8 Nat'l. Bur. of St. [1953]

30i .5 [R(28i )+R(29i )] 9 c.

31i R<'~Oi)go(.05i) 8 c.

I

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·1

I

Obtained y-values were substituted into the forJlD.lla constituting

187

Table gives values of v, good to 8 decimal places, corresponding to

~(v) where ~(v) ma.y equal any integral multiple of .0001 from. 5000

The entry nearest to v , the va.lue of v for which ~(v)oto .9999.

of .001. Thus rather than interpolate in the Bureau of Standards'

Table, inverse reading of Kelley t s Table was done for 1 < Iv I. Kelley's

the Bureau of Standards t Table gives ~(v) only for integral multiples

only when Po = .25, steps 26i to 3li apply only for Zo =e.. 25 •

Steps 22i to 25i yield the y-values used in Simpson's Rule as applied

co + +to f g (t)[~(U )-<P(L )]dt and steps 3li yield the y-va.lues used ino 0

Simpson's Rule as applied to fOg (t)[~(U-)-~(L-)]dt.-00 0

was desired, was found in the body of the table, and then the inverse

of this entry was taken as ~(v ). lu+ I > 1 when and only when i > 5,o

as reflected in the change from steps 10i to s.teps lli and the

corresponding change from steps 14i to 15i • IL+ I > 1 when and only

when 23 ~ i, as is evidenced in the change from steps l2i to l3i

, and

the corresponding change from steps 16i to 17i •

We see that steps 1 to 21. are done once and then hold for allJ.

Zo treated. However steps 22i to 25i va.ry with Zo as go(t) depends

on zoo steps 10i to 25i deal with calculation of the y-vallles for

finding f 00 g (t)[~(U+)-~(L+)dt , While steps 26. to 31. concerno 0 J. J.o

calculation of the y-va.lues for finding f~ go(t)[~(U-)-~(L-)]dt.

Since it was found necessary to evaluate fOg (t)[~(U-)-~(L-)]dt~ 0

Simpson's Rule, which was then evaluated. Where Po = .25, the results

from the two different applications of Simpson t s Rule involved were

added. Fina.11y since each of the ensuing values for E(W(Zo;B:N» was

I

I.IIIIIIII_IIIIIIII·I

188

jUdged to be good to within haJ.f a unit in the second decimal. place,

these values were rounded to two decimal. places as shown in Table 3.2.

Now we consider calculation of E(W(Zo;~»' by means of

Simpson's Rule, 'When Po = .5 or Zo = o. In this case, we regard

- - + +u , L , U , and L . as functions of T23

say, Student's t with 23

degrees of freedom.. Also let t be the d'UJllDy variable standing for

values of T23 and let f(t) be the p.d.f. of T23

• Then the symmetry

of the situation permits us to write

E(W(O;~» fO 2f(t) [tb(U-)-tb(L-) ]dt ,-eo

where, using the notation of Table 7.1,

1u- = (23)-2 R(9)t + R(5) ,

1L = (23)-2R(8)t - R(5) •

O:>taining (23) - i to 7 d. p. from Comrie [1956], performing the opera-

tions indicated above, and thenrounding as can be seen, we have

u· = .1359t + .44136,

L- = .2746t - .44136.

u· and L- were computed to 4 d.p. for values of t at intervals of .2

from -4 to o. All values of U- were rounded to 4 d.p. as were all

values of L- greater than -1. The remaining values of L- were

rounded to 3 d.p. Then for each value of t treated, the areas under

the standard normal curve within a distance U- and L- around the origin

were found in National Bureau of Standards [1953] and rounded to 4

d.p. For t greater than -3.4, the sum of these two results served

as 2[tb(U-)-tb(L-)]. For all other values of t, the absolute value of

the difference between these two results served as 2[tb(U-)-tb(L-)].

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Rule, which yielded

t treated and rounded to 4 d.p. to act as the y-value in Simpson's

of t at intervals of .2 from -4 to 0, in the following manner.

[Taylor, 1955, p. 65l] ,

•

•r(12) = II ~

=

r(12)/r(23/2) =

1.

( /)l.3.5.7 ---·2l·1t2

r 23 2 = II2

in the p. d. f. of T. \ole have that

32(l4.l726) =i5 (l4.l726)

= .94484 •

It remained to find the multiplicative constant r(12)/.J231t r(23/2)

and

each multiplication involved. Finally the 8th power was multiplied

by the 4th power to yield (l+t2/23)12 , 6 d.p. being retained in the

result, and the reciprocal of this result to 4 d.p. served as the

kernel of f(t).

Then 2(l+t2/23)-12[~(U-)-~(L-)] was computed for each value of

powers by successive squaring, 6 d.p. being retained in the result of

(l+t2/23) was found to 6 d.p. and then found to the 2nd, 4th and 8th

l89Next (l+t2/23)-12, the kernel of f(t), was computed for values

Thus

1

= 5. 94572 1t~

Also .J23 1t = (4.79583 )(3.l4l59)

= l5.o6653 •

Hence r(12)/J231t r(23/2) = (5.94572)/(l5.06653)

= .39463

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190

M1ltiplying the result obtained from application of Simpson 's

Rule by the multiplicative constant in f(t), we obtain

E(W(O;~» = (.39463)(.94484)

= .37 ,

as shown in Table 3.2 •

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