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Page 1: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

AD-AC94 732 AIR FORCE INST OF TECH WRISHT-PATTERSON AFB OH SCHOO--ETC F/6 12/1MODIFIED KOLMOGOROV-SMIRNOV TEST FOR THE GAMMA AND WEIBULL DI--(TC (U)

UNCLASSIFIED AFIT/GOR/MA/80D-1

Ehnmmhhhhhhhhhu

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SFEB 9S'

UNITED STATES AIR FORCEAIR UNIVERSITY

AIR FORCE INSTITUTE OF TECHNOLOGYWright-Patterson Air Force Base,Ohio

C.2 _ _ _ _ _ _ _ _

Li.I

Aploed io publi rdU

812 0902

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2 3 JAN 1981APPROVEM) 1""7 F,'L-SE AFR 190-17.(Ii) LAUREL A. LAMPELA, 2Lt, USA

Deputy Director, Public Affairs

V': AFB, 01i .45433 -

~A MVODIFIED KOLMOQOROV- SMIRNOVTEST FOR THE GAMMA AND WEIBULL

DISTRIBUTION WITH UNKNOWNLOCATION AND SCALE PARAMETERS.

THESIS,

// AFIT/Gd'R/MA/8,OD-l / Ramon Crtes2Lt USAF

Approved for public release; distribution unlimited.

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AFIT/GOR/MA/80D-1

A MODIFIED KOLMOGOROV-SMIRNOV

TEST FOR THE GAMMA AND WEIBULL

DISTRIBUTION WITH UNKNOWN

LOCATION AND SCALE PARAMETERS

THESIS

Presented to the Faculty of the School of Engineering

of the Air Force Institute of Technology

Air University

in Partial Fulfillment of the

Requirements for the Degree of

Master of Science

by

Ram6n Cortes

2Lt USAF

Graduate Operations Research

December 1980

Approved for public release; distribution unlimited.

-,si

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Preface

The purpose of this thesis was to generate tables used

for the modified Kolmogorov-Smirnov statistic. These tables

are used for testing whether a set of observations is from a

Weibull (Gamma) population when the scale and location param-

eters are not specified but must be estimated from the sam-

ple. A brief Monte Carlo investigation is made of the power

of the test.

I would like to thank my advisor, Capt. Brian Woodruff,

who offered me considerable guidance throughout my thesis

project.

I would also like to thank my readers, Lt. Col. James

Dunne and Dr. Albert H. Moore, whose advise aided me greatly.

Finally, I wish to acknowledge my gratitude to my class-

mates for their encouragement when the going got rough.

Ramon Cortes

t -:

7 T / .:4" ~ :o i,

IL O

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Contents

PagePreface.................... .. .. .... ... .. .. . . ...

List of Figures. ..................... v

List of Tables .. .................... vii

Abstract. ......................... ix

I . Introduction. .................... 1

Kolmogorov-Smirnov Test... .. ...........Chi-square vs. Kolmogorov-Smirnov Test 2Problem .. ..................... 3Background. .................... 4Objectives. .................... 5

Ii. The Distributions .. ................. 6

Weibull Distribution ................ 6Historical Notes. ................ 6Application of the Weibull Distribution ... 6The Weibull Probability Function .. ....... 7

Gamma Distribution. ................ 8Historical Notes. ................ 8Applications of the Gamma Distribution . .. 9

III. Methodology. ..................... 11

*Generation of Random Weibull Deviates .. ...... 13Generation of Random Gamma Deviates .. ....... 13Maximum Likelihood Estimation of Weibull andGamma Parameters .................. 13Weibull Maximum Likelihood Estimates. ....... 14Gamma Maximum Likelihood Estimates. ........ 15Calculation of Test Statistic D .. ........ 16Power Comparison .................. 18Analysis of Critical Values vs. Shape Parameter 18

.-4 Computer Programs. ................. 19

IV. Use of Tables. .................... 20

Example for Weibull Distribution. ........ 21Example of the Gamma Distribution .. ....... 22

iii

E41 -

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V. Discussion of Results ... ............. ... 25

Presentation of the Kolmogorov-Smirnov Tables 25Validity of Computer Programs .......... ... 26Power of the Test .... .............. 27Relationship between Critical Values and ShapeParameter ...... .................. ... 32

VI. Conclusions and Recommendations .......... ... 35

Conclusions ..... ................. 35Recommendations for Further Study. ...... ... 35

Bibliography ....... ..................... ... 37

Appendix A: Tables for the Gamma Distribution . . . . 39

Appendix B: Tables for the Weibull Distribution . . . 48

Appendix C: Plots for the Relationship between ShapeParameters and Critical Values ...... ... 58

Appendix D: Computer Programs ... ............ .. 89

Vita ......... ......................... . i. 100

iv

I.

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List of Figures

Figure Page

1. CDF and PDF for Weibull and Gamma Distribution. 8

2. Summary of the Procedure ... ............. ... 12

3. Gamma Level=.20 N=5 .... ............... .. 59

4. Gamma Level=.20 N=15 .... ............... ... 60

5. Gamma Level=.20 N=30 .... ............... ... 61

6. Gamma Level=.15 N=5 ....... ............... 62

7. Gamma Level=.15 N=15 ....... ............... 63

8. Gamma Level=.15 N=30 ..... ............... ... 64

9. Gamma Level=.10 N=5 .... ............... .. 65

10. Gamma Level=.10 N=I5 ..... ............... 66

11. Gamma Level=.10 N=30 .... ............... ... 67

12. Gamma Level=.05 N=5 .... ............... ... 68

13. Gamma Level=.05 N=15 ....... ............... 69

14. Gamma Level=.05 N=30 .... ............... ... 70

15. Gamma Level=.01 N=5 .... ............... .. 71

16. Gamma Level=.01 N=15 ....... ............... 72

17. Gamma Level=.01 N=30 .... ............... ... 73

18. Weibull Level=.20 N=5 .... .............. .. 74

:1 19. Weibull Level=.20 N=15 ....... .............. 75

20. Weibull Level=.20 N=30 .... .............. .. 76

21. Weibull Level=.15 N=5 .... .............. .. 77

22. Weibull Level=.15 N=15 .............. 78

V

I,

p]

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Figure Page

23. Weibull Level=.15 N=30 .... .............. ... 79

24. Weibull Level=.10 N=5 ..... .............. .. 80

25. Weibull Level=.10 N=15 .... .............. ... 81

26. Weibull Level=.1O N=30 .... .............. ... 82

27. Weibull Level=.05 N=5 ..... .............. .. 83

28. Weibull Level=.05 N=15 .... .............. ... 84

29. Weibull Level=.05 N=30 .... .............. ... 85

30. Weibull Level=.01 N-5 ..... .............. .. 86

31. Weibull Level=.01 N=15 .... .............. ... 87

32. Weibull Level=.01 N=30 .... .............. ... 88

vi

4l

-- 'I

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Pr

List of Tables

Table Page

I. Results of Example 1 .... ............. .. 22

II. Results of Example 2 .... ............. .. 24

III. Comparison between Thesis Program CriticalValues and Lilliefors Critical Values ........ 28

IV. Power of the Test for the Weibull ......... ... 28

V. Power of the Test for the Gamma .......... ... 29

VI. Coefficients and R2 Values for the Relation-ship between Critical Values and Shape Param-eters ........ ..................... ... 34

VII. Shape Parameter Equal to 0.5 .. ......... ... 40

VIII. Shape Parameter Equal to 1.0 .. ......... ... 41

IX. Shape Parameter Equal to 1.5 .. ......... ... 42

X. Shape Parameter Equal to 2.0 .. ......... ... 43

XI. Shape Parameter Equal to 2.5 .. ......... ... 44

XII. Shape Parameter Equal to 3.0 .. ......... ... 45

XIII. Shape Parameter Equal to 3.5 .. ......... ... 46

XIV. Shape Parameter Equal to 4.0 .. ......... ... 47

XV. Shape Parameter Equal to 0.5 .. ......... ... 49

XVI. Shape Parameter Equal to 1.0 .. ......... ... 50

XVII. Shape Parameter Equal to 1.5 .. ......... ... 51

" XVIII. Shape Parameter Equal to 2.0 .. ........ ... 52

XIX. Shape Parameter Equal to 2.5 .. ........ .. 53

XX. Shape Parameter Equal to 3.0 .. ......... ... 54

XXI. Shape Parameter Equal to 3.5 .. ......... ... 55

1vii

oQ

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Table Page!~lT Shape Parameter Equal to 4.0.... .. .. ...

XXIII. Shape Parameter Equal to 1.0 (5,000) . . . . 57

viii

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AFIT/GOR/MA/80D-1

Abstract

-Z-" The Kolmogorov-Smirnov tables for the Weibull and Gamma

distribution were generated in this thesis. These tables

are used when testing whether a set of observations are from

a Weibull (Gamma) distribution in which the location and

scale parameters must be estimated from the sample.

A power investigation of the test against some selected

distributions using Monte Carlo techniques was conducted.

This procedure shows that the test is reasonably powerful

for a number of alternative distributions.

A relationship between the critical values and the

shape parameters was investigated for the Weibull and Gamma

distributions. No apparent relationship was found for the

Gamma distribution. In contrast, an approximate log-linear

relationship was found for the Weibull when the shape param-

eter is between one and four.<

ix

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A MODIFIED KOLMOGOROV-SMIRNOVTEST FOR THE GAMMA AND WEIBULL

DISTRIBUTION WITH UNKNOWNLOCATION AND SCALE PARAMETERS

I. Introduction

One of the most important sets of data generated by the

Air Force is data dealing with the time to failure of equip-

ment. Failure distributions are of extreme importance to the

Air Force since they determine the reliability of almost all

mechanical systems in service.

Often these time-to-failure data are especially useful

if we can determine to what theoretical distribution they

correspond. That is, a test is carried out to determine an

agreement between the distribution of a set of sample values

and a theoretical distribution. This test is known as the

"goodness of fit test." If the frequency distribution of the

data compares well with the expected or theoretical distri-

bution, we can then use the theoretical distribution to rep-

resent the parent or underlying population. Two very useful

theoretical distributions that deal with time-to-failure data

are the Gamma and Weibull distributions.

Kolmogorov-Smirnov Test

The Kolmogorov-Smirnov statistic provides a means of test-

ing whether a set of sample values are from some completely

PI

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specified continuous theoretical distribution, FO(x)

Suppose that a population is thought to have some specified

cumulative distribution function, say F0 (x) . Then, for

any specified value of x , the value of FO(x) is the

proportion of individuals in the population having measure-

ments less than or equal to x . The cumulative step-func-

tion of a random sample of N observations is expected to

be fairly close to this specified distribution function. If

it is not ose enough, this is evidence that the hypothet-

ical distribution is not the correct one.

If FO(x) is the population cumulative distribution

and S (x.) the observed cumulative step-function of ansample (i.e. S (x )_ , where K is the number of obser-

vations less than or equal to x ), then the sampling

distribution of

D = MAXIFo(x) - S (x)j (i)n

is known, and is independent of F0 (x) , if F0 (x) is con-

tinuous. A standard table of the Kolmogorov-Smirnov test

(1:425) gives certain critical points of the distribution

of D for various samples sizes.

Chi-Square vs. Kolmogorov-Smirnov Test

Another alternative to carry out the tests of goodness

of fit, especially useful in the case where population pa-

rameters must be estimated, is the Chi-Square test (X 2 )

'I

•2

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The Kolmogorov-Smirnov test (K-S), however, has three major

advantages over the Chi-Square (12:68).

1. The K-S test can be used with small sample sizes,

where the validity of the Chi-Square test would be question-

able.

2. Often the K-S test appears to be a more powerful

test than the Chi-Square test for any sample size.

3. The K-S test will usually require less computation

than X 2 This is especially true when a graphical test

is used, for if the hypothesis is rejected, the computation

stops at the point of rejection.

The two major advantages the Chi-Square has over the

K-S test are (12:68):

1. In cases where parameters must be estimated from

the sample, the X2 test is easily modified by reducing

the number of degrees of freedom. The K-S test has no

such known modification, so it is not applicable in such

cases.

2. The K-S test cannot be applied to discrete popu-

lations, whereas X 2 can be.

Problem

The standard tables used for the Kolmogorov-Smirnov

test are valid only when testing whether a set of observa-

tions are from a completely specified continuous distribu-

tion. If one or more parameters must be estimated from the

3

A

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sample then the tables are no longer valid. It has been

suggested by (12:68) that if the K-S test is used in this

case, the results will be conservative in the sense that

the probability of a type I error will be smaller than as

given by tables of Kolmogorov-Smirnov statistics. That is,

the actual significance level would be lower than that

given by the standard tables.

The existence of the probability integral transforma-

tion permits us to generate K-S type tables for any partic-

distribution, if the parameters to be estimated are param-

eters of scale or location. David and Johnson (4:182)

have shown that these parameters will be completely inde-

pendent of the distribution of any test statistic based on

the cumulative distribution function. The distribution of

the test statistic will, in general, depend on the function-

al form of the distribution of the original variables.

Since the K-S type test statistic is one based on the cum-

ulative distribution function, the results of David and

Johnson apply in this case.

Background

Tables for the Kolmogorov-Smirnov test when population

parameters are unknown have been generated by Hubert W..4

Lilliefors at the George Washington University. He gener-

ated the tables for the Normal distribution (8:399) and the

Exponential distribution (9:387).

4

I

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Objectives

This thesis has the following objectives:

1. To generate and present the Kolmogorov-Smirnov

tables for the Weibull and Gamma distribution when the scale

and location parameters are not specified.

2. To determine the power of the test in selected

situations using a Monte Carlo investigation.

3. To investigate the relationship between the criti-

cal values and the shape parameter for the Weibull and Gamma

distributions.

4 5

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II. The Distributions

Weibull Distribution

Historical Notes. The Weibull distribution is named

after the Swedish scientist Waloddi Weibull, who published

a paper in 1939 (10:50) giving some of the distribution's

uses for an analysis of the breaking strengths of solids.

However, it was also known to R.A. Fisher and L.H.C. Tippett

who published a paper in 1928.

Application of the Weibull Distribution. In the past

ten to fifteen years the Weibull distribution has emerged

as the one popular parametric family of failure distribu-

tions. Its applicability to a wide variety of failure sit-

uations was discussed by Weibull. For example, it has been

used to describe vacuum-tube and ball-bearing failures.

The Weibull distribution is closely related to the ex-

ponential, but has two additional parameters, the shape

parameter and the location parameter. Thus, instead of a

single constant failure rate X , as in the exponential

case, a variety of hazard situations can be treated. For

a given Weibull distribution, the failure rate can be

either continually increasing, continually decreasing, or

else constant. Since many failures encountered in practice,

especially those pertaining to nonelectric parts, show an.1

increasing failure rate (due to deterioration or wear) the

o,

•6

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Weibull distribution is useful in describing failure pat-

tern of this type.

The Weibull Probability Function. Let the random var-

iable x denote testing or operating time, and let n de-

note the "scale" parameter, a the "shape" parameter, and

y the "location" parameter. The Weibull probability den-

sity function (p.d.f.) is

f(xlnay) 6(x-y) a-1 exp -( ,i<x (2)

The Weibull distribution function is

F(x) =ff(xH,,Y,)

0

1 exp [,Xy (3)

When a=1 the Weibull distribution specializes to the

exponential distribution, and when a=2 the resulting dis-

*tribution is known as the Rayleigh distribution. The mean

and variance of the Weibull are given as follows:

', Mean = Y+nr(.1) (4)

tN

'1 7

,I

- -

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I~ JI..........

Probability CumulativeDensity Function f(x) Distribution Function F(x)

S 4- I5 .

Vaiac ja2L) r 2-5

I -x

Figure 1. CDF and PDF for Weibull and Gamma Distribution

A Weibull distribution with yO and n-l is illus-

trated for different values of the shape parameter a in

Figure 1.

Gamma Distribution

Historical Notes. The Gamma distribution is a natural

V extension of the exponential distribution and has sometimes

been considered as a model in life-test problems. It can

be derived by considering the time to the Kth success in

Ca Poisson process or, equivalently, by considering the K

fold convolution (10:55) of an exponential distribution.

8A'

i

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iS

The Gamma distribution is the continuous analog of the

negative binomial distribution (11:125) which can also be

obtained by considering the sum of K variables with a

common geometric distribution.

Applications of the Gamma Distribution. The Gamma

Distribution has been used widely in queueing systems. For

example, it has been used to determine the lengths of time

between malfunctions for aircraft engine and the lengths of

time between arrivals at a bank, super market, or mainten-

ance checkout queue.

The Gamma distribution is one of the most useful con-

tinuous distributions available to the simulation analyst.

If the variables from some random phenomenon cannot assume

negative values and generally follow a unimodal distribu-

tion, then the chances are excellent that a member of the

gamma family can adequately model the phenomenon. The

Gamma distribution is defined by three parameters, ,

a and c , where 9 is the "scale" parameter, a is

the "shape" parameter and c is the "location" parameter.

The Gamma probability density function (p.d.f.) is

f(xlcea) = (x-c exp - (x (6)r1(a)8

Oa>O x2czo

9

A

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The Gamma distribution function is

x

F(x) = ff(xc,6,a) (7)

0

This integral does not have a closed form expression.

Its values have to be determined by numerical calculations.

When a=1 , c=O the Gamma density is the exponential

density function with an expected value of 6 . If a

is an integer value, K , then the Gamma distribution is

commonly referred to as an Erlang-K distribution. Further-

more, if 6=1 , then as a becomes large, the Gamma dis-

tribution approaches the normal distribution. If we set

a=v/2 and 6=2 (where v is the degrees of freedom) we

get the Chi-Square distribution. The mean and variance of

the Gamma are given as follows:

Mean = a6+c (8)

Variance = a62 (9)

A Gamma distribution with c=O and e=1 is illus-

trated for different values of the shape parameters a in

Figure 1.

'1

I10L4

•I

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III. Methodology

The method that follows was the one used in calculating

critical values for the modified D statistic when the

scale and location parameter are not specified.

For a fixed sample size n , random deviates x

x(2)...x(n) were generated from the Weibull and Gamma dis-

tributions. Next the random sample x(1 ),x( 2 )... X(n) was

used to estimate the scale and location parameter by the

method of maximum-likelihood. The resulting estimates of

the scale and location parameter and the constant value of

the shape parameter are then used to determine F0 (x), the

hypothesized distribution function. Finally, D=maxlFo(x)-

S n(x) was calculated for the sample size n . This pro-

cedure was repeated 1000 times, thus generating 1000 inde-

pendent values of the D statistics. These 1000 values

were then ranked, and the 80th, 85th, 90th, 95th and 99th

percentiles were found. This entire process was performed

for sample sizes from n=4 to 30

After tables for the test procedure were completed, a

power comparison was conducted.4

What follows is a more detailed description of the

steps taken in this procedure. Figure 2 illustrates this

procedure.

, 11

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GeneratingP. Random Variates

from Gamma (Weibull) Step 1Distribution

Determiningthe scale & location

parameters by Step 2Method of Maximum

likelihood

Obtaining

D~maxIF(x)'S n(X) I Step 3

Go To Step 3

(1000) T

Determinethe 80th, 85th,90th Step 4

95th,99th percentile

END

Figure 2. Summary of the Procedure.

01

'7 12

'-I

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Generation of Random Weibull Deviates

The Weibull pseudo-random deviates were generated by

1

x = I-ln (u)Ia (10)

where u is a pseudo-random deviate from a uniform (0,I)

distribution and B is the shape parameter. These deviates

were obtained on the CDC 6600 computer by using the Inter-

national Mathematical and Statistics Library (IMSL) sub-

routine GGWIB (19:306).

Generation of Random Gamma Deviates

The Gamma pseudo-random deviates were generated using

rejection methods by various computational algorithms de-

pending on the value of the shape parameter a. These

algorithms are contained in the IMSL subroutine GGAMR

(14:1541). This subroutine was used to obtain the Gamma

deviates on the CDC 6600 computer.

Maximum Likelihood Estimation of

Weibull and Gamma Parameters

The procedure used to derive the maximum likelihood es-

timates, n , 6 and y , of the Weibull parameters, r

3 and y , and e , a and c ,.of the Gamma parameters,

e , a and c , was developed by Harter and Moore (7:639).

They developed an iterative procedure for censored or un-

censored samples for the three parameter Weibull and Gamma

13

4

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distribution. For each of these distributions, the likeli-

hood function is written down, and the three maximum-likeli-

hood equations are obtained. In each case, simultaneous

solution of these three equations would yield joint maxi-

mum-likelihood estimators for the three parameters. The

iterative procedures proposed to solve the equations are

applicable in the most general case, in which all three

parameters are unknown, and also to special cases in which

any one or any two of the parameters are unknown.

Weibull Maximum Likelihood Estimates

The probability density function of the random vari-

able x having a Weibull distribution with location param-

eter y O , scale parameter n , and shape parameter B

is given by

f(xlnB,¥) = ((x-Y ) B-1 ep-I -

n B

n,3>0, xzy>O (11)

The natural logarithm of the likelihood function of the

order statistics xl, x2 P ... Px of a sample of size n is

given by (7:640)

w n

L = ln n!+n(ln $-a ln n)+(6-1) ln (x.-y)C=1

n

,- (Xl-Y)/n (12)

c=1

14

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The maximum-likelihood equations are obtained by equat-

ing to zero the partial derivatives of L with respect to

each of the three parameters; these partial derivatives are

given by:

S 6n n (1 (13)- +S (xi-y) /n

i=l

n nIL _ ln (Xi-y)-E (xi-y)/n] ln [(xi-Y)n (14)

i=1

n n3L (I-a)E (xi-Y) +BniL(xi-y) (15)

i=l i=l

Gamma Maximum Likelihood Estimates

The probability density function of the random vari-

able x having a Gamma distribution with location param-

eter c>O , scale parameter a , and shape parameter a

is given by

fxc6c)= [llr(a)6J [xc/J~~x[(~)6

O,a>O x>-caO (16)

The natural logarithm of the likelihood function of the

order statistics xl,X 2 ... xn of a sample of size n is

given by

15

.1

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L in n!-n In F(A)-na In @+(a-l)

n n

• In (xi-c)- (xi-c)/6 (17)i~l i=l

The maximum-likelihood equations are obtained by equat-

ing to zero the partial derivatives of L with respect to

each of the three parameters; these partial derivatives are

given by

n

IL = -nca/O + __, (xi-c)8 2 (18)i~l

n

@L = -n In O + in (xi-c)-nr'(c)/r(a) (19)

i=1

n

Te-1-=(l-cx) (.c) /e (20)

3L

where the prime in 771 indicates differentiation with re-

spect to a

Calculation of Test Statistic DFor the Weibull distribution there was no problem in

-4

calculating D in Eq(1) since F0 (x) has a closed form ex-

pression Eq.(3). Since the Gamma distribution does not have

a closed form expression, an integral calculation had to be

16t..

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done in order to determine FO(x) before calculating D

This integral calculation was accomplished using the IMSL

subroutine MDGAM (16:946) which evaluates the probability

that a random variable from a Gamma distribution having

shape parameter a with c=O and 0=1 is less than or

equal to x

x

I(x,) = f e-tta-ldt (21)' ~r(a) (1

0

In order to use this subroutine, the generalized three

parameter Gamma density function considered in this thesis

had to be transformed to the one parameter Gamma density

function used in the subroutine. This transformation was

done as follows:

y X-- (C)

F(y) f (x-c)- e dx (22)

C

Let, z=x-c dz=dx.

Then, x=c-z=O

X=Y-z=y-c

andy-c z l -z

F(y) = f z e dz (23)l~0 7 (a )e7

Next, let t= , dt -7-

17

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Then z=O-t=O

z=y-c*t-y-c

and

e a - 1

F(y) = f et dt (24)

0

which is the same integral in Eq(21) with x=

Power Comparison

The power of the test using the modified D statistic

was compared for four alternative distributions. These

power comparisons were made using Monte Carlo simulation.

The procedure is very similar to the procedure used in creat-

ing the critical values for the D statistic. Five thou-

sand random samples of size n were generated for each of

the four alternative distributions considered. Then the D

statistic was calculated and compared to its respective cri-

tical value obtained from the table generated. The number

of rejections of the two different null hypotheses (that

the distribution was from a Weibull or Gamma distribution

with known shape parameter) were counted.

Analysis of Critical Values vs. Shape Parameter

j The tables generated in this thesis for the Weibull

and Gamma distributions will depend on the value of the shape

parameter. For different values of the shape parameter we

will have different tables. For this reason we

ac 18i.

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have presented tables for eight different shape parameters.

These shape parameters are .5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5,

4.0

Since there is an apparent relationship between the

shape parameter and the table's critical values,an investi-

gation of the possibility of finding an expression or equa-

tion that indicates the relationship between the shape

parameter and the critical values was conducted. Both re-

gression and graphical methods were considered.

Computer Programs

The computer programs built for generating the Gamma

and Weibull distribution were made very flexible. That is

they were built to generate tables for any shape parameter

input into the programs. The programs used to generate the

Gamma and Weibull distributions are presented in Appendix D.

1

'! 19

.4

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IV. Use of Tables

In the Kolmogorov Smirnov test a theoretical or as-

sumed distribution F(x), is compared to an observed

distribution S (x) If the maximum deviation of

IF(x)-Sn(x)I exceeds a certain limiting value , d n

the assumed distribution is rejected.

The steps in applying the modified Kolmogorov Smir-

nov test when the scale and location parameters are es-

timated are as follows:

1. Determine the values of the scale and location

parameters by one of the methods of parameter estimation.

2. Specify completely (including the shape param-

eter) the hypothesized distribution F(x)

3. Determine the desired level of significance, a

and the desired or planned sample size, n . The level of

significance, a , is the risk of rejecting the hypothe-

sized distribution if it is in fact the true distribution.

4. Using the tables generated in this thesis, select

the critical value, da,n

5. Select a random sample of n items from the pop-

ulation to be tested, and order the observations.

6. Determine the maximum value, d , of

D = IFo(x)-S (x)n

20

I,

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where

Sn= i/n when x ix<xi+ 1

7. If d>d reject the hypothesized distribution.

If dSd the hypothesized distribution cannot be re-a,n

jected and we say that it is reasonable to assume that the

hypothesized distribution is the true distribution.

These steps are illustrated in the two examples which

follow.

Example for the Weibull Distribution

A certain engineer is involved with the development of

a new turbojet engine. It is important to determine the

failure time of this new engine before it is introduced to

the market. Six identical engines were tested. The failure

time data obtained from these tests were:

.034, .168, .266, .563, 1.344, 3.118 years.

The engineer believes that the failure time of this engine

follows a Weibull distribution with shape parameter equal

to one and scale and location parameter undetermined. We

will now conduct the test developed in this thesis at the

5% level of significance to determine if the engineer's

,! hypothesis is reasonable. That is,

H : The distribution is Weibull (shape=l)

H : Another distributiona

First we determine the scale and location parameter by

21

'r

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TABLE I

T

i x. F(xi) Sn(xi) IF(xi)-Sn(xi) I IF(xi)-Sn(xi-llI

1 .034 .036 .167 .131 .036

2 .168 .166 .334 .168 .001

3 .266 .250 .501 *.251* .084

4 .563 .457 .668 .211 .044

5 1.3441.767 .835 .068 .099

6 3 .118 1.966 1.000 .034 .131

Note that d=maxjF(x)-Sn (x) is .251.in

the method of maximum likelihood. The following values

were obtained: scale = n =.923 location = a= .0003

Using these estimates and 8=1 , in Eq(3) we can obtain

values for the hypothesized distribution. We can then con-

* struct Table I. Using the table of critical values for the

Weibull distribution for shape=1 we find for a=.05 and

n=6 that d.0 5 ,6 =.383 Hence, since .251<.383, we can-

not reject the null hypothesis. We conclude that a Weibull

distribution with shape parameter equal to one is a reason-

-4 able model for the engine failure data.

Example of the Gamma Distribution

Major Smith is the maintenance squadron commander at

22

A

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717 '

Big AFB. During the past six months he has been trying to

determine the lengths of time his crew takes to complete a

maintenance service for an aircraft engine. This informa-

tion is vital to him since it will permit him to schedule

his crew in a way that will optimize service time. Major

Smith, presented this problem to Lt Jones. Lt Jones gath-

ered the following service time data:

.397, .524, .691, .973, 2.548, 2.933 hours.

Lt Jones told Major Smith that he believes that these data

follow a Gamma distribution with shape parameter equal to

one. Major Smith wants to use the test procedures of this

thesis at the 5% level of significance to determine if Lt

Jones' hypothesis is reasonable. That is,

H : The distribution is Gamma (shape=l)0

H Another distributiona

First we determine the scale and location parameter by

the method of maximum likelihood. The following values

were obtained: scale = 6 = 1.009 location = c = .341

4, Using these estimates, and a=i, in Eq(24), we can obtain

values for the hypothesized distribution function. We can

then construct Table II. Using the table of critical val-

ues for the Gamma distribution for shape=l we find for level

an=.05 and n=6 that d 0 .383 . Hence, since .220<.383,.05,6

we cannot reject the null hypothesis. We conclude that a

Gamma distribution with shape parameter equal to 1 is a

reasonable model for the length of time of a maintenance

service. 23

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TABLE II

i x i F(x1 Sn(Xi IF(xi )-Sn(x) I IF(xi)-Sn(xi-l)l

1 .397 .054 .167 .113 .054

2 .524 .167 .334 .167 .000

3 .691 .293 .501 .208 .041

4 .973 .466 .668 .202 .035

5 2.548 .888 .835 .053 *.220*

6 2.933 .923 1.000 .077 .088

Note that d=maxlF(x)-Sn(x)I is .220.

I

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V. Discussion of Results

The results obtained in this thesis are discussed in

this chapter. Results applicable to each objective are

presented in the sequence in which the objectives were pre-

sented in Chapter I. All tables referenced in this chapter

are located in Appendix A and B, and all figures in Appen-

dix C.

Presentation of the Kolmogorov-Smirnov Tables

The Kolmogorov-Smirnov tables for the Weibull and

Gamma distribution when the scale and location parameters

are not specified are presented in Appendix A and B respec-

tively. The critical values in these tables were subject

to simulation variability. This variability was reflected

in the third decimal place of the critical values. Due to

this variability the critical values in many tables are not

monotonically decreasing functions of sample size when the

sample size exceeds 15.

This variability is not unusual for any kind of Monte

Carlo simulation. There are several methods which can be

used to eliminate this variability. One method is to in-.4

crease the simulation sample size. In this thesis the

critical values were obtained with 1000 samples. If the sim-

ulation would have been done with more than 1000 samples, the

25

1~ .

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variability would have been less. For example Table

(Ref. Appendix B) was obtained with 5,000 samples and as

can be seen the simulation variability has been reduced.

The critical values are monotonically decreasing for in-

creasing sample sizes at all levels of significance. All

the tables were not generated with 5,000 samples because to

do so would require a large amount of computer time. Anoth-

er method to reduce simulation variability is to run the pro-

gram with a different seed several times with a relatively

small number of samples in each run. Then the critical val-

ues are obtained by computing the means of the critical val-

ues found on these several runs. For example, the programs

used in calculating the tables in this thesis (1000 samples)

could have been run five times with a different seed every-

time. Then the critical values would have been obtained by

computing the mean of the five program runs. Again this

procedure was not carried out because it, too, would require

a large amount of computer time.

Validity of Computer Programs

The validity of the computer programs (Appendix D) used

to generate the tables presented in this thesis were verified

Vby comparing the critical values obtained with the computer

programs with the critical values obtained by Lilliefors in

his exponential table (9:387). This comparison is possible

because both the Weibull and Gamma distributions become an

26

-''D

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exponential distribution when the shape parameter is one

and the location parameter is zero. When the shape param-

eter was input to one, the location parameter to zero and

the scale parameter set to be estimated in the computer

programs the critical values obtained were almost identi-

cal to the critical values obtained by Lilliefors in his

exponential table. These results can be seen in Table III

for sample size n=10

Power of the Test

The power of the test was carried out as discussed in

Chapter III with only one exception. The Monte Carlo sim-

ulation for the power of the test for the Gamma distribution

was obtained with 2000 samples of size n=30 instead of the

3,000 samples indicated in Chapter III. The reason for this

change was the excessive amount of computer time required to

do the test with 5,000 samples.

The results obtained are presented in Table IV and V.

Table IV represents the probability of rejecting the null

hypothesis of a Weibull distribution with shape parameter

one using the modified D statistic of this thesis whe n

the sample size is 30. The numbers are the result of Monte

.4 Carlo simulation with 5,000 samples for each distribution.

Table V represents the probability of rejecting the null

hypothesis of a Gamma distribution with shape parameter one'I

using the modified D statistic when the sample size is 30.

27

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TABLE III

Comparison Between Thesis Program Critical Values

and Lilliefors Critical Values

Level ofSignificance .20 .15 .10 .05 .01

Critical ValuesObtain by .263 .277 .295 .325 .38Lilliefors

Critical ValuesObtain by .265 .279 .301 .323 .381Thesis Programs

TABLE IV

Power of the Test for the Weibull

Modified K-S test for the Weibull

distribuonDistribution Using critical values from Table 23

n = 30

iq: Weibull(shape=1) a'!"5 u, P2 0=.01 Pi u2

geibull .05 .044 .056 .01 .007 .0133eta .98 .976 .984 .94 .933 .947lormal .96 .955 .965 .86 .850 .870

og Normal .41 .396 .424 .18 .169 .191amma(shape=1) .05 .044 .056 .01 .007 .013amma(shape=2) .46 .446 .474 .26 .248 .272eibull(shape=2) .05 .044 .056 .01 .007 .013

28

a

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TABLE V

Power of the Test for the Gamma

Modified K-S test for the Gammadistribution

UnderlyingDistribution Using critical values from Table 8

n = 30

Ho: Gamma(shape=l) .05 P1 W2 O..0l W1 112

Gamma .04 .031 .049 .01 .006 .014Beta .68 .660 .700 .34 .319 .361Normal .42 .398 .442 .26 .241 .279

Log Normal .10 .087 .113 .04 .031 .049Weibull(shape=l) .04 .031 .049 .01 .006 .014Weibull(shape=2) .01 .006 .014 .001 .000 .002

The numbers are the result of Monte Carlo simulation with

2,000 samples for each distribution.

In both Tables IV and V, U, and U2 represent the

lower and upper limit of the 95% confidence interval for the

probability of rejecting the null hypothesis. This confi-

dence interval is given by:

P(P-Z P<P+ZP(U 1 <P<U2 )

where p is the proportion of rejections in the computer

runs and Za/2 is the upper a/2 cut-off value from a

standard normal.

Table IV indicates that when the null hypothesis is

29

'4

-' *1

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true, the test does in fact achieve the claimed significance

level. The test has a very high power when used on Beta and

Normal distributions, but not so high for the log normal dis-

tribution. The results obtained for the Gamma distribution

with shape parameter equal to one were the same as those for

the Weibull with shape equal to one. This result was expect-

ed since both distributions are an exponential distribution

under this condition. In contrast, when the test was done

with a Gamma distribution with shape parameter equal to two

different results were obtained. This demonstrates that the

test does reasonably well against the Gamma distribution

with shape parameter not equal to one. In contrast, the

test is not very powerful against a Weibull with shape par-

ameter equal to two. This suggests that the test may not be

able to readily distinguish between different members of the

Weibull family.

Table V indicates that when the null hypothesis is true,

the test approximates reasonably well the claimed .05 signi-

ficance level and exactly the .01 significance level. The

* test has a reasonably good power when used on Beta and Nor-

mal distributions, but no so good for the log normal distri-

bution. The results obtained for the Weibull distributionV with shape parameter equal to one were the same as those for

the Gamma with shape equal to one. This result was expect-

ed since both distributions are an exponential distribution

93

'130

.!.

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under this condition. In contrast, when the test was done

with a Weibull distribution with shape parameter equal to

two, different results were obtained. Unfortunately, the

results obtained for the Weibull distribution with shape

parameter equal to two were poor. They indicate that the

test could not distinguish between a sample generated from

a Gamma distribution with shape parameter equal to one to

a sample from a Weibull distribution with a shape parameter

equal to two.

In general, the results obtained for the power of the

test for the Gamma distribution were poor in comparison to

the results obtained for the Weibull distribution even

though the null hypothesis in each table was the exponen-

tial distribution.

Three possible factors could have caused these differ-

ent results for the Gamma.

1) The critical values used in the Gamma test were

generated with 2,000 samples while the critical values used

in the Weibull distribution test were generated with 5,000

samples. This implies that the critical values for the

Gamma distribution had more Monte Carlo variability thanI,

.4 those for the Weibull distribution.

2) The power of the test for the Gamma distribution

was conducted with 2,000 samples while the test for the

Weibull distribution was done with 5,000 samples.

31

r

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3) The cumulative distribution function for the Gamma

was determined by an integral calculation. This integral

calculation is subject to errors since the values obtained

will be limited by some tolerance. For the Weibull distri-

bution such a problem does not exist since its cumulative

distribution function has a closed form.

Relationship Between Critical Values and Shape Parameter

As discussed in Chapter III a relationship between the

critical values and the shape parameter was investigated.

For the Gamma distribution no apparent relationship was

found as can be seen in Figures three through 17. Possibly

by increasing the number of shape parameters in the inves-

tigation a relationship could be found. Another possible

explanation for the apparent lack of a relationship is Monte

Carlo variability in the critical values. Increasing the

number of samples used to generate the critical values for

the Gamma distribution may reduce the variability sufficient-

ly to enable a relationship to be found.

For the Weibull distribution an approximate relation-

ship was found when the shape parameter is greater than one

as can be seen in Figures 18 through 32. The approximated

equation found to represent the relationship between the

critical values and the shape parameter (between one and four)

is

ln y = a 0 + aix

32

I.

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14

where ao is a constant, and a, is the coefficient of

the independent variable. The variable x is the indepen-

dent variable which is the shape parameter, and in y rep-

resents the natural logarithm of the critical value.

This equation was the best expression found to repre-

sent the relationship between the critical values and the

shape parameter between one and four.

The equations obtained for all levels of significance

using three different sample sizes are presented in Table

VI.

The R2 value on the table indicates how good the log-

linear equation represents the relationship between the cri-

tical values and the shape parameter. From these values we

can see that with the exception of sample size 15 at level

of significance .01 all R2 values are greater than .85

This indicates that the log-linear equation is a good approx-

imation for the relationship. For example, sample size 15 at

level .20 had R2=.95, and sample size 15 at level .15 had

R2=.96 . In fact, of the 15 regressions done, 10 had an R2

equal to or greater than .90. From these results it can be

concluded that the log-linear equation is a good approxima-

tion for the relationship, but not an exact one.

SThe values presented in Table VI were calculated using

the Statistical Package for the Social Sciences (SPSS)(17:1I).

These equations can be used to obtain approximate critical.1

values for Weibull distributions with any shape parameter be-

tween one and four.33

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CAA

.4 N. r co cr (m 0'o r.. 0 M CSJ U, cl

44 -

.0 co CD m D tz

r-4 CIS__ _ __ _

.0 4mmC

- -d

I= 1.4 I I I ko M

V0 0 DC

$4C 4D4 CCC'J 0' CJ

~ 4-4 cl *% *~ *l

---

u

~~ro m) (a _ _

Z It

-of00

'~ .4 0' 34

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VI. Conclusions and Recommendations

Conclusions

On the basis of the results obtained in this thesis

the following conclusions are drawn.

1. The tables generated for the modified Kolmogorov-

Smirnov are valid. Monte Carlo studies have shown that the

test achieved the claimed significance level when the null

hypothesis is true and have good power against several al-

ternative distributions.

2. A log-linear relationship between the critical

values and the shape parameter was found for the Weibull

distribution when the shape parameter is greater than one.

In contrast, there seems to be no simple linear type rela-

tionship for the Gamma distribution.

Recommendations for Further Study

Based on the observations made during this investiga-

tion, the following recommendations are proposed for further

study:

1) To increase the number of shape parameters used in the

investigation of the Weibull (Gamma) distribution to see if

a better expression can be obtained for the relationship be-

* tween critical values and shape parameters.*4

2) To increase the range of shape parameters investigated

35

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for the Gamma distribution to see if an expression or equa-

tion could be found.

3) To increase the range of shape parameters investigated

for the Weibull distribution to see if the log-linear re-

lationship between shape parameters and critical values re-

mains valid for shape parameter greater than four.

4) To compare the power of the test between the Chi-square

and the modified Kolmogorov-Smirnov test for some selected

distributions.

5) To modify the Anderson-Darling and Cram4r-von-Mises

goodness-of-fit tests in a manner similar to the one pre-

sented in this thesis for the K-S test. The result of

the three modified tests can then be compared to determine

which is most powerful.

6) To conduct the power of the test for the Gamma with

5,000 samples to see if the results could be improved with

a reduction in the Monte Carlo variability.

.3

63

t4

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Bibliography

1. Birbaun, Z.W. "Numerical Tabulation of the Distribu-tion of Kolmogorov Statistic for Finite Sample Size",Journal of the American Statistical Association, 47,1952, 425-41.

2. Caplen, R.H. A Practical Approach to Reliability Bus-

iness, Book Limited, London, 1972, 41-51, 1975, 256-62.

3. Carter, ADS, Mechanical Reliability, John Wiley & Sons,New York, 1972, 31-39.

4. David, F.N. and Johnson. "The Probability IntegralTransformation when Parameters are Estimated from theSample", Biometrika, 35, 1948, 182-190.

5. Durbin J. "Kolmogorov-Smirnov tests when Parametersare Estimated with Applications to Tests of Exponential-its and Tests on Spacings" Biometrika,62,1975, 5-22.

6. Green and Bourne, Reliability Technology, John Wiley &Sons, New York, 1972, 244-249, 239, 537.

7. Harter, H.L. and Moore, H.A., "Maximum-likelihood Es-timation of the Parameters of Gamma and Weibull Pop-ulation from Complete and from Censored Samples," Tech-nometrics,7, 1965, 639-643.

8. Lilliefors, H. "On the Kolmogorov-Smirnov Test for Nor-mality which Mean and Variance Unknown" Journal of theAmerican Statistical Association, 62, 1967, 399-402.

9. ------- "On the Kolmogorov-Smirnov Test for the Exponen-tial Distribution with Mean Unknown" Journal of the Am-erican Statistical Association, 64, 1969, 387-399.

10. Locks, M.O. Reliability, Maintainability and Availabil-ity Assessment, Spartan Books, Hayden Book Co., RochellePark, N.J., 1973.

11. M1ann, Schafer, Sinpurwalla, Method for Statistical An-alysis of Reliability and Life Data, John Wiley & Sons,New York, 1974.

9I

12. Massey, F.J. "The Kolmogorov-Smirnov Test for Goodnessof Fit" Journal of the American Statistical Association,46, 1951, 68-78.

37

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13. Offen, W.W., Littell, R.C.,McClaur, J.T. "Goodness ofFit Tests for the Two Parameter Weibull Distribution,"Commun. Statist.-Simula. Computa,B8(3), 1979,257-269.

14. Robinson, D.W. and Lewis P.A.W. Generating Gamma andCauchy Random Variables. An Extension to the NavalPost Graduate School Random Number Package. NPS-72Ro 7541, Naval Postgraduate School, Monterey, Califor-nia, 1975.

15. Shannon, R.E. Systems Simulation the Art and SciencePrentice Hall, Inc. New Jersey, 1975, 64-68, 80-88.

16. Stegun, I.A., Abramowitz, M. Handbook of MathematicalFunctions, Dover Publications, Inc, New York, 1965,946-948.

17. Steinbrenker, K., Nie, N.H., Hull, C.H., Jenkins, G.J.SPSS - Statistical Package for the Social Science,McGraw Hill, Inc., 1970, 11-131.

18. Stuart, A., and Kendall, M.G. The Advance Theory ofStatistics, Vol 2, Hafner Publishing Company, NewYork, 1961, Chapter 30.

19. Wald A. and Mann, U.B., "On the Choice of the Numberof Class Intervals in the Application of the Chi-Square Test" Annuals of Mathematical Statistics,13,1942, 306-17.

-4

38

4,

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Appendix A

Tables for the Gamma Distribution

'339

'4

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.5 i .10 .s .0

4 ... 3 .. ........~... A 3 ._9 . 0

5 . -. ,'50 , 36? __, . ._3 . ...... ._€ ....... ._30 .49?

6 - 326 . . o 34, -

7 ~ ~ CF -.1 .jZ?____

* . . .. -. zT..= ° 263 . 3...3 z - .41?

Ia ._ ,2e5 . _, -

12. - - ,23 ... ...... 2.28 .. . ,3L...0........ .. °33F. ***

12.5. .23 .1 3_._.7_ ,_07 ._,367.__.._.

18 ... -.... 221 , e.. . °O. ,53 0 . o3 0 _

14. . . .- l . . . Z! . _ ______.o3C; .___

16. .... ..... d~ ..195. ,2 1 a ,. _ ," ___ _ . . . 313F1, . 195

3 . ..

18 . .._ e . .1.. .. 5 *2jo _._. __,____5 ._

1. -q .... . *t5 j 88 .23 .234 28.. ZO... . ... tO LIS -_ . z.vO0 . 2t13 .2"s

21 .. .... , . .. .. ,. 01 . . . ? 4 _ ,283___

_ -Z__ . 1 ,X. ... --__L_ ___ _, .181 _-__.- .... _.._.204

( j ze ~~~2 .... .... ...U L _ - .. Z, 6 .t _, .'

"J 29 136 - 2..... .- 1 7 _..1 ____ .0H . .. 28. _ I . ,___.__

Table VII. Shape Parameter Equal to 0.5

40

e" 4. ., -I

Page 53: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

zAMPLE Lh.EL OF SIGNIFICANCE FOP *=ucIA (X -S(M)SIZE ' " " "'

N .20 .15 .10 .05 .31

.9 .35 " .399 .35 .519

- T11 -3 r7 34 .739 449

J -2 r. 4C-. 6" ,383 - . 56--

7 - .261 .2§0 .324. M-56 .4ZI.

S " -- -28'4 .3 . 36 .413

9 - .27. .291 M F- --- "-

i- .- .266 .286 .312 .3 1

11 .iZ5 .25 .272 .g * .Z.. 2b1-r7 .294 Z65

13 .215 .2 3 .2'3 .270 ".329

14. .2C .24...'1 - .263

5 .194 .a0 7 .2-5 -Its.5"

16 .193 Z20 .. 16 .2-2 .303

-- 9 17Z .185 .2 cc .22- .26S

-71. •166 .1-76 .1 3o .216 "."b' " ' -

22 • 1.9 . 169 , 65 .2-o07 ".240.

, 25 - l et .17T-; 1 -6 .. . 4Z ;.E

Z 2 .IS6 .166 L .81 .9 Z,

25 • .. 1o, .4 76 .O Z:

! - ' ~LUT .[5 ML Z .19 -" z3,

15. t

Table VIII. Shape Parameter Equal to 1.0

41

•4

, , .t -

Page 54: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SiZEN .!a .15 G10 .05 .1

6 ... .312--- .. -,330 -- .- 5'--38- - .44--1-....

8 --- 274 .... ... .284 -- .33 5 .336-. .. -. 391- .

9- - - . .2635 . 7253 64.2-92-- - 29 ---. 32 - -

12 .. .-. 213 -- 229 ,2 6- ,,27-- - 327 - -

,.- -- w13 9. .--. *205-- -r24 .260---- ,----

1- "5 .. . . .,2 ,.5- -19 . 46-- -2 65

1 bo .. . ... 193 . 198 .2-18 - .2 -39 - -. 30 --

- 17 - ,1 -- -- 193 - 13 6.2 36 --. 26 -3 -

L ...9 - 253 z69 --

2 - --... 16 -. .. --- ------------- --. - - -- 2 7-

L2iI 71 1?Tal X SaePraee qult .

-242

-27/

-2 139

- 2-Z - . . I5 . . .. t 6;. 6. 9 . .. . .. 9-

-3

Table IX. Shape Parameter Equal to 1.5

t4

Page 55: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

a .M .E -QE OF 51ibNIFICANCIE FOR O:4A.X(v 00-5(4)SIZE

-- N , -.2 ... . ..-- ,i .. ... uS .... . ..3 1- ...

.4 .361 .377 .'.CJ. .436 .490

5 •33, .351 .369 .4;7 .46,.

.3G] .31b .339 .370 ,*,5

7 *8.3j.32.4 .161 .4.11

7 .287 .26J .3 A .336 .3El

2. .2b. .264 .311 .357

I, .24.) .25. .272 .2 9d .34

11 ,2?:. .243 .263 .286 .3,38

1Z .215 .22w .2 -*6 .269 .317

13 2Z., .2Zv .2 36 .2 60 .310

Iv196 .209 .222 .244' .2c0

10 .18: .19 .213 .23* .250

17 L15 M; ;201 .226 .268

Fi .174. .185 .198 ,Z25 .Z8

19 ,17J .181 .195 .212 .265

24 .163 17f .186 .205 .251.

21 .1j •17-J .181 .233 .235

2z •.16 .171 .L64 .26 .229

23 .15. 1bL .177 195 o238

"" 2... •1 157 .171 ,J39 .23a

25 .14i5 .151 .168 .185 .220

2b .144 .15. .164 .185 .240

27 .141 .11) .164. .183 .2c5

20 .141 .15.b. .1 8'L6 .2?'.

.,29 . 133) •146 ,.15b 1 7,3 .222

3d .135 .142 .155 .?2 .2C5

Table X. Shape Parameter Equal to 2.0

'1 43

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JSAPIOLE LEVEL OF SITFICANCE FOI (=MAX(F(X)-S(X))

F- I .21 .5 .10 .05 .01

r- ...... __.. ~ ....... 3?s _ ____ . .. _ -. . ..

. 6 -_ __ 3C3 __ ,.____.__'3.3 _ .35 . .. *30t _._

7 __ 478 2~j -- .3991

':__+____ ,264 47 , ,3 ,ZT.... .37 _ __.. ,..L0_._.. .__...Z.2 E• 0 . ,-q.. .... ____O- .__ ._99 25 . -.0 t.264 *_Z! % "01 . . . ... 3€ ..2

__1_.. .. 42I________ 31 _____2.5 .___ * 268 . .. ,313 __

,._t _.___ . _ _ . .... .J3 . .

r 21.1 .3' .29 0

16 , ,?0 .j41 3 ,2; __ 277.

__ ._ .. .. _1, . 21.

. J ._ _ _..183 _. %9h 220 0 -........ .. 2 !7

-- . ,J S . 1!5t_'. _ ,. 1 _ _... 2 15 __

-0 163 .. ______ 1.51_,__ 1 . .. 2 0, _2Q

Zl .*16 1_6. . tL ... ..A S .27 . "..O _

..._.._ 2• J-FQ ,L_ .. ...j. *. ... .._ . ,. _____ .2 0..

IAI

_ .- 173 . , .2 2

2, ..14 - "5 172

Table XI. Shape Parameter Equal to 2.5

44

Page 57: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SAMPLE LEVEL OF SIGkIFICANCE FOQ CzMAXF(XI-S(X))

.420 .15 .10 .05 0

35 - 6__ *0?-36 -. 4 9 ~32I -. 31 3--.-36- . !91--- -. 4% -•301 ,w35 ,31 - -,362 - . ,a -- '

---- .261 .261 .291 .26 -. 376-

.23 .26-- -- 3 ?6

.1" -7 201 .216 .• 35----- .275---

7- - .- 1 1 8e .•206 .226 266-

---- " .-- 3 -. 218- - .2 . 43

- - 6" 6 . 2, L-- - - -. 2 .

22- .~157 -6.-- 7

2.3- -150 .1oe- .3 ..

-24- .1oj ;16e .172 .15 --i- -160

-29--8I-- 136---s e-- °---- o5207 * 2 C I

- -F1 .13913 0-

Table XII. Shape Parameter Equal to 3.0

14 45

7''- -'--------3 -1'-I • 8 - . 9..

Page 58: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SA4PLF LEVEL OF SIGNIFTCANCE FOR CAX(F(X)-S(XIl-__- -- -- - -. *..-.... ... - - -- -~-

4 20 . s .10 .05 .of

=; .37?. . . ,.3 e 3 S,_,3,4 7 .. ..45q ___6 zg ,32 3

1A ,.0.Z.. . .4Q1...........88 ___

3,22 .331 ., 1= . 3 *1t_ _ .q . .._ 6 .299 .1

-_ ,26 f, . .2. *3 . . 373

. 9 .2 3 ,257 . .273 .30 . ... 3(9_

10__ .L . ....... 2 9 .2127 .... 1. .27.5_____ .3 9

1- 211 - .-2.-0 .. 2 2 8 . 2 .307

7.,. . , 2238 .2753___69 _ ,_ ,___ .3LO. L ......

2 ! t .. ....275 ..

J7. .,. . 7 . .. 1 ,.0 ,__6__A.2811___0 _. ___ 2 5

.. __ ___ . 1 2~ _, __ .1 1 -------- .- .-2-'.

_. ..1? 1.80 .216______ ,25

.__Z ... .. 66 iTi _ .__ 8L . , 0 _____. . .. 22 _ _

__zZ, .r.,;_._ tzo . ._ _ _ _ __ _i.. .21 . .. ... . .•153 __ __ '_ ..- 165 _ .--11 . . .2 6 --

. .. _ __ _ ,143 9 . . .14_ 6_ 1. 5 |6; e ._ 2 IS._• ' i 7 . .. .t9... .,5 ___ __~

° , .17____2- ..

______..3 __ *X . LS , J17______ _ __

__ ___.+ __ t1,13.64 _ ___. • ......... . l1 , iZJZ.. .7 *..139 ~150 at.1a

3 t________ 7_____211 _

4t33 i.o119 0

.130 .13F .1i Iqq

Table XIII. Shape Parameter Equal to 3.5

46

Page 59: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SAMPLE LEVEL OF SIrNTITCANCE FOQ CzMAX(F(XI-StX))

L.25 .10 .05 .01

. . . . . . . . . . . .. __"_______.. .. 2 "__'_ _ __.. .. .

j .,j3?_______.3:3j ,._.2...........,35 82__ __

6 __, *26 ,__ _ ,310 .,, _.7._, 8 .. ..

_ ____ _ , ______ 3 ________ 3. .3336 .385

S.32 .383 ....

S .23. ,2 ,0_ - .269 3 29S _ , .3".

0. •_*23 ' ,2' 7 ,z_,_ _ .__ .362,

- -_ _ *217 .. . ,,_____ ? *.2 .______ *271 .... .3C5

12 -___ ,10 ___ 21__ ..... . .256 .... 3(

. 15, ,0 , 6J. 3 17 . _ _ .28 _._.

ZA' Z 413 .229 2? .7I _ a ... _____ ,___,_., . . .. _

- -_..._... . . .... J6L............I3.LIZ_______.1q....... ,2 ;'_...

I.I

- 't

22 ..,1. 5__ .z. .2 .. 4 ,1 7,t9 2 ....

•~~1 V56_3 _ 1_t 62 L?_. 0 . .. 2;5_

"' ~ ~ ~ 2 ----- - L39 ... . 14 k 7_. _ 17, e _7 .. ... zq .

"- 7 _ ._ - 9__ 1 3 , 14J ,,_5 . . . 2 9 .

Z9 -- 91.,, Z. . . LS3 .. .o 07- 11... . 196

~~~30 .f32 .t~.,l n.I. l.

Table X'IV. Shape Parameter Equal to 4.0

'! 47

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Appendix B

Tables for the Weibull Distribution

'1

48

I4

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Z MP E .EVEL OF S FIGNIFICANCE FOR CL&4AX{F(X)-S |)SIZEN .20 .15 .10 .05 .01

-. 35- -- 45I-5 ... "-----.337----..... .353 . ....-. 368 :0 -5 -

6 .316 .333 .350 .383 .453

- -- .... ---- - 309 -- 326 356

--- 9---.- -.... 26- -. 279- -. 219----.320 .v~ -

.236- --. 252- - .26--.-290-.354-

-13-- .334-

--- "i---222- --. 236 - 2502 3 "-

-- -1 -. . 21- -- -.222-- -239 .26 5- - r- -

.. 16-- .2Z- -a214 .229-- - 3

--- . .194-.---- - .203- - -. 2 17 35- .- "'

-- 19 - ---- 1-- 310 - -" 9 -2 -. 2 • -3.2 68---1 3 . . .. •177-1, 2 9 . 3 e 5 --

-to

2-- ---- .1s1 9 t.--.1 3- 11 . . . .. '13.n3. . . 19V "*2t4 - .r2 --'-

"- 26- -- --------- w2 129 L

-- 29-- --- 1 -- -- 187- 17. ;t .19 ,2-36- - -

Table XV. Shape Parameter Equal to 0.5

49

--"4- - ' 7 -+1 9" 1 9 - -9 t • 0 ' --- -...)- '- fL7 - --- 8 t 9 e 0

Page 62: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

LEVE OF SIGJZ1FCA.NCE FOR Dz.iAX(F(X)-z XI)SZZ £ .. .. . . . . . . . .

.. ,10 .05 .01

.... ... . - ,339 ... ..... - 359 ..... .. 3 99---- - -4 35

" -- -... 330 . . 362 - .19gc r44..9--....

-5. .2'. -- .301 - 3 26--- ---- i357

-"1- --. -.226-------2 28.--- -33 6-38---

2 61- ----. 2-7- .239-.3 • 34-

-14'-- -v-24-- -. 23S ---. 2449.27'

....15q -- - --- --- 2ir •2 20- 2 39 t65- - .--31.

- "16-- ---- 201O -- 212 --- 2 29 - 2 2 3C3-

- 17 .... ---- "19?-------209 -. 2 26 - r2d - 3C -

- -8 -- ---- ,199- -. 21 2

3 19--- --v 193 .20 3 . . .. 2 16 1237 -5------ 290 -+-- Z . .. 85 "19b do 12"2 Ta9- z?

- - - -- 7'a 6------e0 .- 2- ' --- ,I1 2-32,26.

23---- ---.... . "-' ---. ", --- -w.191- 1'.0 -250

--- - -9--i.-" ---- ±5- 16,0236------

159- 24'I ~ ~~~---'8 ... ..--- 9,19-10 2 0'.3 -

- 21- -. 156 --- -163 - 1 73 - .g 7- 1

Table XVI. Shape Parameter Equal to 1.0

50IS

Page 63: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SAM.PLE LEVEL OF SIGNIFICANCE FOR DOlAX(F(X)-F(),SIZE . .. . . . . . . . ,7 N4 , o15 •10 ,q •cl

I. *373 0392 o417 ktr6 oS52

5 0341 .356 0.02 .421 o4F8

6 .329 .345 .361 S-F9 o473

e *299 .315 o336 o767 .432

a # 285 . 0. .318 0369 *413

9 .27 -284 •1 .326 0395

1S2 .274 .293 .319 .37?

11 .251 02& o283 311 0345

12 .23' .2S4 e274 .298 *359

13 *2.q *239 .251 .2?1 .329

14 0 221 .23:i .248 .274 32

15 .21tt .222 4'39 .ZF9 .315

16 .21! .213 .230 .253 03t1

17 .230 *212 .227 029 03;2

18 .131 .209 212 0231 o2f.9

19 o134 .203 .215 ,2?9 .206

20 $186 .196 o208 o229 .2e9

1 21 01st f189 0207 o225 ,26522 0176 .186 .196 9214 .247

23 o1?'. .18'4 .195 24052. .17t2 .182 .19.0 .210 .243

2. *139 .177 .190 .218 .253

26 o16, *174 ,184 .282 .237

27 .162 .170 6185 •2"1 ,2 1S28 .sq .1?1 .182 o?12 .235

29 .155 .163 ,173 .186 s233

-22

Table XVII. Shape Parameter Equal to 1.5

51

*t~~

Page 64: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

LLVEa OF Slk;NZFICANCt FOR Oz4A,((F(X)-SUK))

N '0 1 .1a 0 _______

6 .319 .332 .391 .379 .457

*7.287 - .30 1 r .59

10 - - .255 -2 3

-- 1-.-2n6 247 .932U~

23 -; - '333 715r

14 19- Z-13 - 299~

19 232 .±8 Z±9s r-

-- ;11 .2 C.e'IT

.5 Tan .15 .125b

'.16

52

Page 65: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

A> ,,;-:;" I. CF S I G1, FIC. N; E FC R 01AX(F X)-S ) )s ze

N .11 .15 . 340 .05 .01

. .- .5 7-

• 3.32'- .3' 5 .176 .,5

... $ ... ... 27 . -2qF" 3 -3 .3W .;r3

7 -. , .... '2*24 - .Z6 -, .2" 63 Z89 -7714 -

3 W) 2 .23 Z7. - '

-5 " "m2T3 .21- .z9 . b

16 ;19T .z

17 " TM f'93 . .2-f4 .. 38 .[9-

.- -- -- -"'--;-, 9z 2z 5 . .7 b * z bb

jg:) ........... --- ;±q'- .2;6 .. e3. { -

'- - - . . 175 .igF•72 • T•-

SZ ~ -; 170 i'17 1- .c'o .,ui

- 2. 167 17T flab 3U

.... -... .... .5' .168 .lbb .*L ., 0-b- -

• , Table XiX. Shape Parameter Equal to 2.5

5,53

. • L "

Page 66: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

L Ik. L OF SIGNIFICA.JC- FOR j="1iA.F(X3-S())

*s:z§N .23 .15 .10 .05 .02.

3' . 7 .3 . 39

Z91 19.414

. 272- .2 7 '1 or--- "--

. 1- .. . 236 . 252-- .2"8 0 -

... 12-. :" ;2~. 3( ,6 ".e + 3 ---

" "- .1- 2 8lt 2 Z+ .. . . 0

.2±27 Z #4 Zbb .

i7-21 ,179; .20 *zil ["€ Y Il

18 *6 183' * 19"7 •2 :.. *r I 9 ., ,

.-- T L72 . a 1 2 zTr

21Ig - 73 217.l r lii.i ~ .

2 -7rl----7r-. *1 * g

- , . - ..9 2 ZS .2 .27 -

.161.19

I~ ;.7 .56 .1(5j

Table XX. Shvwpe Parameter Equal to 3.0

54

Page 67: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

I MPLE .1/ri. OF SIG-41F ICAN;E FOR J3II Xi(X(),-S(J3

N .3. .1.5 00 05 .31

.30369 . 12 - 1&1#5.92

.32 .34.2...56 .397, • n,: e321 - T? fl2-: ,' ; :

7 .281 .295 .31s .3147 .611

.26) .2' .300 .337 .. 02

.5 - 269 2 A6 .6

IL 21., .261 .278 ,37

11 .27, Y. .2 26 PA3 7

12 7. ;1.23 -2 A.. ,2 - I 7 -137

13 .21 22u .2472 .1_64 .3gpl

148 21 .27 .71.5 .794

6 5 .-*1 a 3 . 219 24.0 .261.

l1, A z.. .216 .-Y6 .292

is . 181 .' . -- ,- - . 2 . 63

9- .184 .192 .203 .218 .271

20.17;* 1&7 .199 .'Zi .260

2t . 171 .18- ,196 .217 .25

,22 .e I.176 1 da ._n5 62 i!

.16. .175 188 .. 6f.

2s .. 17 .183 sc .2Pi1

25.161 .16 .b •181 .20J3.;)_

26 1 .s1 .166 .178 .192 2:!l

275.14 .1-94- .2y?

'21 .16•, .173 .192 .225

29 ,1 .7 .155 .. 63 -270

.1467 15. *163 .182 .234

Table XXI. Shape Parameter Equal to 3.5

55

Page 68: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

iLF'.'L I SINNTCTCANCE PCQ r=MAX(.

N ' '2 .15 .10 .' .0.1

6.304. 3?0 _ .. 342 .-3 E9 -. 4'3A

7 . 271 ., .2.. ..6 .34. .. .414

. .26 1 .243 .300 .335 .4(2

9 -- 5 . 2 ,7 _ . .. .:255 .310 .3f?

to . . . _ _ _ .2 . . .277 ..306 . 7

11 .2!T .24e _ __ .265 .24? 33

12 .223 .214. .259 ?85 .3117

13 . 215 .26 .21. . q .262 .3IO. .

14t .209 .2!1 _ 2"36 1.258 .298

14 .tq8 .208 _ ,2-? .2,4S .296

1. .. .. .i t . ....... . ....... ,.218 .239 .20 _.

1? A1.9 .201 .213 .23E .290

t4 .181 ....... 148 . ... .. .220 .260

. ... 3 .112 .. 2 2 18 .74 ___

20 .173 . 185 ,190 - ,221 .?60

21 .072 .131 .195 .216 .258,

2 11 166 .176 .1AE .?12 .26

21 .164 .174 .187 .204 .2&4

24 . . 13 . 172 . .1q3 .200 .2.9

. _ . . . .16 .179 201 22

26 .t57 .166 .177 191 . .229

?1 _ .155 .16g. .173 .192 .2,15

24 "_ .150 . 160 . 173 11 .225

A 29 *1'.7 .154. .162 .179 .227

30 .146 .152 .162 .182 . 214

Table XXII. Shape Parameter Equal to 4.09&

56

*r

Page 69: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

AMPLE L. 4E OF SIGNFI ,- N^- FOR D=AX (F(X)eS{)

SIZE.20 .15 .to .C5 .01

33 ,364 .393 .30 .496

. 31] .325 .348 .383 .444

7 .29 .300 o 30 .363 , ,g

8 - .276 .4113~ .'.

9 *261 .275 .292 .- o .375

10 .25. ,266 .283 .309 .365

.2 .23 .2. i .35

13 .225 .23' .277 .32'

1.0 .22 .231 .2.t. .T-

S). 221 3 6 .23 T.314.

Lb 2C7 .21 .3 .253 .303

17 .26212 .226 *o

i - .195 .207 .Z20 .?2z .2F -

19 .19L .201. .215 .238 .262

20 .e 197 ,U tz ] i'

21 .181 .191 .204'.

22 .175 .187 .2'O .19 .256

34 .7. 181 .13 .211 -251

--- 5 .1F7 .;1"7 7 . aiO -----

.16 26T • 7'1 • 1 .*71 .0 • 2

28 . i 157 .166 .178 .197 *z 8

Table XXIII. Shape Parameter Equal to 1.0 (5,000)

57

'-I

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Appendix C

Plots for the Relationship between

Shape Parameters and Critical Values

58

4

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SHAPE VS CRITICAL VALUES

GAMMA

LEVEL-. 20 N4-5

U)

~0

CI)

1 0-.8 .6 2'4 -0 40SHP AAEE A

Fiue3 Gma.ve=2 =

L59

Page 72: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

GAMMA

9LEVEL-.20 N-15

Uix

U

06

-4

-JS--j.

I--

(-

.U

0* 2

o 0g . 80 1.60 2.40 3.20 4.0SHAPE PARAMETER VALUES

• , Figure 4. Gamma Level=.20 N=15

60

'I

*1

Page 73: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

GAMMA'a

LEVEL-. 20 N-30X

-4

-

w

_J.<

Uc)

00,1 0:8 1:o 24 .0 40

SHP PAAEE VAUSI .-61

tU

-4!

.89 1.59o 2. 40 3.2a 4.00o, SHAPE PARAMETER VALUES

i"' Figure 5. Gamma Level=.20 N=30

'1 6

Page 74: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

GAMMA

LEVEL-. 15 N-5

~cw

U,

cd.

00 of

==Ma

Page 75: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

GAMMA

F uLEVEl. 15 N-15

-63

0"

'4 63

4-I

' ° • o

Page 76: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS cRITICAL VALUES

GA MMA

f. LEVEI'L-. 15 N-~30

N

U'

U)

N

0,

64

Page 77: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

GAMMAin

0 LEVEL-. 10 N-5

.a

w

<In

>.S 1.0 2.0 320 40

SHP AAETRVLEFiue9.Gea ee.1 =

'1X

65

Page 78: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

GAMMA

ILEVEL-. 10 N-15

CC)

0"

w<m

(I> :m

LiJ

.4

.-.

U

Figure 10. Gamma Level=.1O N=15

'61

' L n s: e 11 e 2166

SHPePRMEE VLE

Page 79: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

GAMMA

LEVEL-. 10 N-30

-4

Z4

wo4

InX-

1:0 :0 8 :0 32 40

SHP PAAMTR ALEFiue1.-4 Lvl.0N3

I67

Page 80: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

GAMMA

MLEVEL-. 05 N-5

tt-

'-4

CY

'00 0.80 1.50 2.40 3.20 4.00I SHAPE PARAMETER VALUES

Figure 12. Gamma Level=.05 N=5

68

Page 81: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

GAMMA

LEVELU. 05 N-15

co

to

w

-J

U, l

.i4

N..060 s 1.50o 2.40 3.20 4.00SHAPE PARAMETER VALUES

Figure 13. Gamma Level=.05 N=15

69

40

Page 82: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

GAMMA

LEVEL-. 05 N-.3-4

w4

-o

D

g.."U

-)U

c0

4-

00 0:B0 1.60 2.40 3. 20 4.00SHAPE PARAMETER VALUES

Figure 14. Gamma Levels.05 N=30

70

.4

Page 83: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

GAMMA

CY LEVEL-. 01 N-,5

-I

In

w

I

,In

i r

._71

-j,

-J

.4

V1

•. 080 1. 6 2. 40 3. 20 4.00SHAPE PARAMETER VALUES

• 1 Figure 15. Gamma Level=.QOl N=5

~71

A . ,

Page 84: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

GAMMA

ILEVEL-.01 N-15

-

w

-i

<1> cc

-1

"M

.80 1.'60 2.40 20 4.00SHAPE PARAMETER VALUES

Figure 16. Gamma Level=.O1 Ni15

72

4

Page 85: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

T-1

SHAPE VS CRITICAL VALUES

GAMMA

M LEVEL-.01 N-30

-

w

.J

-U

"CY

U' tN

\4D

.4

0. 80 1.60 2.40 3.20 4.0SHAPE PARAMETER VALUES

Figure 17. Gamma Level=.O1 N=30

'1

y ° .,.__ 73

Page 86: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

WEIBULL

NwLEVEL-.20 N-5WI

m-

44

w

-j

"M

U

C"

'a

ob:oo 0:80 1:80 2.40 3:20 4.00SHAPE PARAMETER 'ALUES

Figure 18. Weibull Level=.20 N=5

74

Page 87: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

WEIBULL

LEVEL-. 20 N-IS

4m

w

-

U

U

00 0.80 1.60 2.40 3.20 4.00SHAPE PARAMETER VALUES

Figure 19. Weibull Level=.20 Nin15

75

Page 88: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

-- J

SHAPE VS CRITICAL VALUES

WEIBULL

Fg e 0 bLEVEL. .2 N-33

ini

-0 g . 8 1 . B 2 2 . 40 3 .2 09 4. 0 0SHAPE PARAMETER VALUES

i.L Figure 20. Weibull Level=..0 N-30

76

Ii

Page 89: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

WEIBULL

ILEVEL-.15 N-5

'40"

0"

'

.

w

-J

U

U)

(4

00 1.80 2.40 3.20 4.0

SHAPE PARAMETER VALUES

U Figure 21. Weibull Level=.15 N=5

77

Page 90: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHiAPE VS CRITICAL VALUES

WdEI OULL

(4 LEVEL'.. 15 M-"15

(4m

(4

-1

0-.

-4j

SHAPE PirAlTER VAUES

Figure 22.' Weibull Level=.15 N=15

78

Page 91: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

E IULL

LEV/EL-. 15 ,a3

-4to

WI

.1

I.3

Ci

9d

: : Figure 23. Weibull Leve l.15 N=30

< m

U1' J .. ...... I -- "II I III III I II I IIII1 .. . ..

a)I

Page 92: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

WEIBULL

LEVEL-. 10 N-5

w

~I)

I-

m

U,

ca

>qc.

.. L 80 1.60 2.40 3.20 4.00SHAPE PARAMETER VALUES

Figure 24. Weibull Level.10 N-5

80

- .111 II I i

I II II I I

Page 93: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

WEIBULL

el LEVEL-.0 N-15

MI.

Iw

C-

I-

0.8 1.60 2.40 3.20 4.00SHAE PRAMETER VALUES

Figure 25. Weibull Level=.1O N=15

81

.. .. ... .

Page 94: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

7iiSHAPE VS CRITICAL VALUES

WEIBULL

FLEVEL=.I1 N-30

J82

1'-j

to

Page 95: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

,,- -.- .--

LEVC'.- .5.4

c)iU

U

J -I

UCC)

-

it

Figure 27. Weibull Level=.05 N=5

r 83

Ia

p -I. ..-- -' ..:S- - ,, :-,. .. . . t'U. -

Page 96: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

7-

'C)

C,,

*~j1

L

'I

U

4Figure 28. Weibull Level=.05 N=15

84

Arm

t1LeT

Page 97: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

WEIBULL

0. LEVEL-.05 N-30

, ' I

DI

~.0 0.80 1.60Z 2.40 3.20 4.00SHAPE PARAMETER VALUES

i-

Figure 29. Weibull Level=.05 N=30

OWO

85

'4

Page 98: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

AA9 732 AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OH SCHOO--ETC F/6 12/1

F .T... T AMODIFIED KOLMOGOROV-SMIRNOV TEST FOR THE GAMMA AND WEIGULL 01--ETC (U)UNCLASSIFIED AFIT/GOR/MA/800- 1L

Page 99: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

SHAPE VS CRITICAL VALUES

WEIBULL

LEVEL-.01 N-5

'"

4

.91

<U

tn'

U t I0 /

'Vo go .83 1.60 2.40 320 4.00Z0. 0 SHAPE PARAMETER VALUES

Figure 30. Weibull Level=.O1 N=5

'1

r 86

4p. ,

Page 100: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

io

C

"Co

4-rCl ;~

'C.3.

Figure 31. Weibull Level=.Q1 N=15

87

Page 101: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

C"..

<r i

N

Up.~±1a

cc

Fiue3 . Webl eel .1 N3

I~88

Page 102: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

Appendix D

Computer Programs

89

Page 103: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

LIPS!4 YI ..

CC 3T~? - L: F0.1V Tl4-2 MUtCIFT!o TS Tri!c 7 T'L 1- ! r ; V.'uO FOR; 'H WSIRULL )TSTqZ3JtLON

C *Sri%! IF 3C.ALS RAlllTE1r T4-TA IF KN~OWNC *s: atIF Zr TIr' S TI 3: 7-.lcTLOC I Z g Z F I.~ M- K I~ '('OkIGWC al: fF IS 71 1- ir r MATEI

C ir T^, I( I G 1,14

C 0TI:TZ:;IiL -771iTcl IF T74U4 (OR KhJWN VALOIS?c IC 211 :71'. FITI~ OF , 1, (O- Kko'*J VLUE) :0. Oa ((0C Of~1AT~ 1 a~ ALP14A (0%( KNOWN VALUE~)

0'.1: r bOO.0

MP; TX( 0

IX~ I' J4911 3~'SN' j

68 .fT'IiAf)3 9S0.0 ,10

as0 CCITN'JfTIJ TaO

AA(c~rs91

goo! COciTZNJF

ST .0p

.14 4,E~VcL air SIGodrFCAMCVrFai 4 , .tX e()S (X) I*J (11 -*,) 9/,: Irv

X, 2.i 4,j-~

Page 104: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

Cf ''0 T --. ,)j ii

KL :<SJ

V2

:

32 fr -Ti-je

31 EL

IF(

M-57

35 C OJt2

4 -t ) Siilt37 JJuJ-1

':K2 0.

6 l 6 20-39

7 T'AfTi(j)zrwcrA fIj)GO ~

* IF (p") tl I2

GC Tn 924 x( i ) T4F I (J J)

LS 30

LPaL*IV MLP) 24 WL

53 J.SsLSIt

54 LSUL!L.

IF (LS-L) 1-23iP,,8

10 TJ C1

Go TO l

61l IF (Act-((.)~LI La-)7,3221 CrPITZ':JF73 ' TT-' J J 2 (L 0)

91

Page 105: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

11 2.~ L. (I (11 J I

LL 'LL's L

43 VLSL!IIF (L'.*L) 1- 5

44 V-Lrt

GC T: iC'.7 IF YM )Y L) qZIts LLLL-L

51 CC iIJ'!;2 v J) so V-

12 c (Ji)UC (J )I62 IF (: SS) ?5*21,'

To !r *.,

%cc it

L~L xL - ILP2L.1

IF, C(() 34,24.4039 L3'LS-L

'.3 LeesL S +1

41 Xf(LP)w.5-ffL)G^. T(j Z?

GO TO Z?70 IF M-)*f(L.ll IZo~

71 LL9LL-1

GO T P 1

Page 106: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

23 Cr" J24. CCJ)stCL*)

)) Tr 1.)

69 ;4z.

36 3JLNLL')(CT)1!f))

CIJ ( J 12 9i

393

LI..

Page 107: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

':' E*i TZ:j :1'77j., ,.OFT3, OX4,3CA TYe LIi, 9:'".

C T TL' I~' S~ %-IL* FC; THP4: GaPMA 'lIR1t3JTOli

- F -!0 AL2141) I I i* WN

C ~ ; _0s1 . i~ * rC -Sr-sl IF r" ?O14 ~c Ti - T T1. I ' T n rTm-TA (T: KNOWN VALIIEC IfiiI 3- 1 cO; '(NLWN V-6LUL)SO. 04 Pc,)c *A. = t r: T3 'A T- .L~. GO 4104N V ALIM 0

C 1.4

CC "C.fv (10)

Ott 1. : '+ Z

00 P.3 )Ms10OCALL r:%!t*SrS3, %1PA,' o)

':AL L VSFIT1(PN)

T(! ) &PC T)3 ClJ% I .JF

C..-L !I(V J PS JtA J)37 3 Lzt,1A

CALL 1~(4 V~ O

333 C..%TI*.JE

0^'sea 121 0*

cx 0 T V IIIF (EOr .0?. DIP) Ura.4OT

853 CfrJT1i&C'..L 'I5IT' (ll,l

ot 94

Page 108: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

ST

-y - I ,-v

XIL "L, 3?. 1 T, 9 , ? , 1VLrL71A 9L ,,4E4E,~.

EN )THETL. 6a

C L17 Z 1 IC JJ

C.' - /o t

86 L;MO.30

67 *j'qwt*ti )IC CS'qpT I)J: 620t1,

J tie20fl t TfY:1 3-1 1P

)I meJ~

(36 : M :1J0 t" S ~

20 it*S2 I-st,1!! 1

.3 1

rF(J-JL SIJ p ISt (J,

IF E IC).6,i22,13q

'13

117 J~x95

Page 109: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

'.; 4; C.1

GL - t 1~A 2

T3 I MriAz4( eJ)+C11* QN5±,rr.D27D)#N5

G,; I i . ,t '

. , ,

7' ' -I *

GP.8 :a t If (r j L: 1J LPA)J

2-t1tl )lAIs AL 4) A8 Ll).7 TA (J *C(MRL0 -(J ) *0 .O A (

ZFcKf*() IL9P,19195

6G'- TG 1 1120# T7 a(i I*1 .311U4A()

Go TO t(4105 F (TJ 10 OLT(IfV I r4 13p 1.

10 TO .;-107 TwrTtJ)'rw(() #*LTt~f(?WL4j?TH(KKIICfOLTICV-OLT(C11

13 AL:HfMJ=AL*41r$J)

96

Page 110: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

(3 T1-32 nrt, ' CCI)-i- 1:) 1/T4ET,4J)-LP4A(J))

39 (F S-

I1 TLW ( V ! -< * r6 PI ~ *A f

"G' T i

CI( : (-

GC To. 1181 GI3=H-,A)T%(Tr)( ) -L<)TH<)Tt(K),ALPMA()- 4(

52 C(,TJSC.?16 tC TO ) ) a C( j I9 53 C(JSaCE( .? I

46 Gm,=GAT2 A. : 1 J

83K1x

0; Ff --.,954~K-

sos0.0

Page 111: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

14." 7" ' - '

Sa S:. r 'cSL -O0 0

or- - ~ iI

G% G :: z; (?C ) -C~ J) / To, " A) 9LP4 A J)

le I.- -j/2c T .fI

MzJT4ST -rJ)FJ=PL2 (J3)

tF '-'

J.?)Tt4

63 :FD 1 T -()- .)3-.! S pNSfi F( -- "A T J-74t7 J)1 6 96 t

C IF( C-.4.! il *A.444% J) -1 .3-4 #,4 *4Il

C3 1047.J

4 C*ILvT!NJE ,') CLX,

66 weup.

tr .N3C VF7- .- *as 4 *1 4.-46

GOTI I

RE 7 W

EN

c , ,LGCaC 16a 4 k 4 a s 4 * ,460,4

.4C

c9

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Page 112: AFIT/GOR/MA/80D-1 - apps.dtic.mil · tinuous. A standard table of the Kolmogorov-Smirnov test (1:425) gives certain critical points of the distribution of D for various samples sizes

-': n

C

C M- - ..

LL:-L.2'L.ZLL'I(L ) (- UILU I.Ll 1 40 (',!LL-.C.JCZ (ELL)

TC J F ~C' N FJ17ti TC11GAP aI C M , 7

UJ( z fI.e?#r V! 1/Z

CO I L~~

OLL aLL

I -!zSU+J M)-V (L)or* miitP.-'7 UPFN

99

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VITA

Ram6 n Cortes was born on 23 July 1956 in Caguas, Puerto

Rico to Ram6n and Isabel Cortes. He graduated from Gautier

Benitez High School in 1975 and received a Bachelor of

Science degree in Chemistry and Mathematics from the Univer-

sity of Puerto Rico. He was commissioned a Second Lieutenant

in the United States Air Force through the AFROTC program on

June 13, 1979. Lieutenant Cortes entered the Air Force

Institute of Technology Resident School of Engineering in

pursuit of a Master's degree in Operation Research in June,

1979.

Permanent address: 4-P-1 1st Villa del Rey

Caguas, Puerto Rico

100

.1

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S9Cu IWfi.W;i oF THIS PAGE (*hen D.. Et.ed),

REPORT DOCUMENTATION PAGE READ INSTRUCTIONSREPORT ____ ___ ____ __ _ BEFORE COMPLETING FORM1. REPORT NUMBER Ii. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

AFIT/GOR/MA!8OD-1 0, ' ,4. TITLE (end Subtitle) S. TYPE OF REPORT a PERIOD COVERED

A Modified Kolmogorov-Smirnov Test for MS Thesisthe Gamma and Weibull Distribution withUnknown Location and Scale Parameters e-- 6. PERFORMING OG. REPORT NUMBER

7. AUTHOR(a) S. CONTRACT OR GRANT NUMBER(S)

Ramon Cortes, 2dLt., USAF

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASK

Air Force Institute' of Technology/AFIT-EN AREA & WORK UNIT NUMBERS

Wright-Patterson AFB, Ohio 45433 -

I1. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

December 1980 -13. NUMBER OF PAGES

I4. MONITORING AGENCY NAME & ADDRESS(II different from Controlling Office) IS. SECURITY CLASS. (of this report)

UNCLASSIFIEDIa. OECLASSI FICATON/ DOWNGRAOING

SCHEDULE

16. DISTRIBUTION STATEMENT (of this Report)

Approved for public release; distribution unlimited.

1?. DISTRIBUTION STATEMENT (of the abstrect entered In Block 20, If different from Report)

Ill. SUPPLEMENTARY NOTESApproved for public release; IAW AFR 190-17

Frederick Lynch Major, USAFDirector of Publin Affairs,

IS. KEY WORDS (Continue on reire, eside f nece ary1 and identify by block nmber)

Modified K-S Test Critical ValuesGamma Distribution RegressionWeibull DistributionMonte Carlo Simulation

20. ABSTRACT (Contiue en reverse aide It neceeary and identify by block number)

The Kolmogorov-Smirnov tables for the Weibull and Gamma distri-bution were generated in this thesis. These tables are used whentesting whether a set of observations are from a Weibull (Gamma)

r- distribution in which the location and scale parameters must beestimated from the sample.A power investigation of thetest ag/ainst some selected distri-

butions using Monte Carlo techniques was codducted. This procedu

DO I J AN713 1473 EOITION OF 1 NOV 65 IS OBSOLETE UNCLASSTFD

SECURITY CLASSIFICATION OF THIS PAGE (When ots Entetad

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SCCUAI, 4t.WiFM6 oF THIS PAGE, . , Data ,ret.dE)

shows that the test is reasonably powerful for a number of alterna-tive distributions.A relationship between the critical values and the shape param-

eters was investigated for the Weibull and Gamma distributions. Noapparent relationship was found for the Gamma distribution. Incontrast, an approximate log-linear relationship was found for theWeibull when the shape parameter is between one and four.

I

r~i UNCLARRITn

SECURITY CLASSIFICATION OF TWO PAGEWIIen Data Entered)

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D ATE

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