CHAPTER 4

APPROXIMATIONS TO MOMENTS OF ORDER STATISTICS

4.0. Introduction

In the last chapter we have discussed some universal bounds for the moments of order statistics. In

this chapter we shall develop some methods of approximating these moments of order statistics from an

arbitrary continuous distribution. First of all, we study the order statistics from a uniform distribution in

detail in Section 1 and derive exact and explicit expressions for the single and the product moments. Then,

as fIrst noted by K. Pearson and M. V. Pearson (1931) and later worked out in great detail by David and

Johnson (1954), these moments of uniform order statistics may be used to develop some simple series

approximations for the moments of order statistics from a continuous distribution. (This is to be expected

after all in view of the result that the probability integral transformation U = F(x) transforms the order

statistic Xi:n from any continuous distribution into the order statistic Ui:n from a uniform U(O,I) distribu­

tion (see Chapter 1).) These series approximations are presented in Section 2. Some similar series expan­

sions have been developed for the fIrst and second order moments of order statistics by Clark and Williams

(1958) and these are discussed in Section 3. A different kind of series approximation based on the logistic

distribution rather than on the uniform distribution, due to Plackett (1958), is presented in Section 4. In

Section 5, we discuss the fIndings of Saw (1960) who has employed the Darboux form for the remainder in

a Taylor series expansion in order to obtain bounds for the remainder term when the expansion for the ftrst

single moment J.1i:n (1 ~ i ~ n) is terminated after an even number of terms in the series. A very interes­

ting and somewhat involved method based on orthogonal inverse expansion which gives approximations as

well as bounds for the single and the product moments of order statistics has been developed by Sugiura

(1962, 1964). This method of approximation is discussed in great detail in Section 6. Sugiura's method,

however, requires that the population have a fInite variance. In Section 7, we present a generalization due

to Joshi (1969) which provides similar bounds and approximations with less stringent conditions. By noting

that these methods do not give satisfactory results for extreme order statistics, Joshi and Balakrishnan (1983)

have devised a method by which Sugiura's bounds and approximations may be sharpened considerably,

particularly for large sample sizes. This method of improvement for the extreme order statistics is

explained in Section 8 and some comparisons of various methods are also made.

4.1. Uniform order statistics and moments

Let Ul' U2, ... , Un be a random sample of size n from the uniform U(O,I) distribution with pdf f(u) =

1, 0 ~ u ~ ·1, and cdf F(u) = u, 0 ~ u ~ l. Then the density function of the i'th order statistic Ui:n (1 ~ i ~

n) is given by (eq. (1.7»

B. C. Arnold et al., Relations, Bounds and Approximations for Order Statistics

© Springer-Verlag Berlin Heidelberg 1989

74

n! i-I n-i f.. (u)=(_I)'(_)'u (I-u) , O~u~I 1.n 1. n 1 •

from which we get

k n! JI k i-I n-i E(Ui :n) = (1-1)!(n-l)! 0 u u (I-u) du

_ B(k+i,n-i+l) - B ( 1 ,n-1+ I) ,

where B(a,b) = r(a) r(b)/f'(a+b), (a,b > 0), is the complete beta function. Upon simplification, we have

E(Uk )- i(i+I) ... (i+k-I) I""'" k>1 i:n - (n+l)(n+2) ... (n+k), ;:::. 1 ;:::. n, _. (4.1)

Setting k = 1 in (4.1), we have

E(Ui:n) = n : l' 1 :5; i :5; n. (4.2) Keeping in mind the probability integral transformation discussed in Chapter I, we may interpret the result

in (4.2) as that the set of n order statistics divide the total area under the curve y = f(x) into n + I parts,

each part with an expected value V(n+I).

Denoting 'lOW i/(n+l) by Pi and 1 - Pi by qi' we have from (4.1) and (4.2) that for I ~ i ~ n

E(Ui:n) = Pi and Var(Ui:n) = piq/(n+2). Similarly, consider the joint density function of Ui:n and Uj:n, I ~ i < j ~ n, given by (eq. (l.ll»

_ n! i-I' ,i-i-I n-j fij:n(u,v) - (1-1)!G-l-1)!(n-J)! u (V-UT (I-v) , 0 ~ u < v ~ 1.

By considering the transformation

WI = U. /U. and W2 = U. l:n J:n J:n for which the Jacobian is w2' we get the joint density of WI and W 2 from (4.4) as

_ n! i-I -i-i-I j-l n-j f(wl'w2) - (1-1)!G-l-1)!(n-J)! WI (I-WIT w2 (l-w2) ,

(4.3)

(4.4)

o ~ WI ~ 1, 0 ~ w2 ~ 1. (4.5)

It is easy to see from (4.5) that WI = Ui:~j:n and W2 = Uj:n are statistically independent. Moreover, we

realize that the marginal distribution of WI is Beta (ij-i) and that of W 2 is Beta (j,n-j+I), which is in

accordance with the result in Chapter 1. Using this property of independence of WI and W 2' we may now

find

and hence

Cov(Ui:n,Uj :n) = piq/(n+2). (4.6)

Similarly, starting with the joint density function of Ui 'n (r = 1,2, .. j, 1 :5; il < i2 < ... < iJ. ~ n) and r'

then making the transformation W = U .. /U. ., r = 1,2, ... ,j - I, and W. = U .. , we shall be able to r lr.n lr+ I·n J Ij"n

show that the random variables WI' W 2"'" Wj are statistically independent and also that the marginal

distribution of Wr is Beta (ir,ir+Cir) for r = 1,2, ... j-l and that of Wj is Beta (ij,n-ij+I). Making use of this

independence, we may find the product moment of a general order as

75

j-1 I· f k. .J . = J] B lr + e=l e, lr+l - lr _B~I ......... ..----:::-~....,.-''h-....L. l r ,l r+1 -1r

r=

n! fr I [ir - 1 + e!/e]! J [n + e!/e]! t=t [ir-1 + :~:ke]! . (4.7)

The flrst four cumulants and cross-<:umulants of uniform order statistics have been expressed in terms of p. 1

and qi by David and Johnson (1954). These quantities will be used repeatedly in the following sections in

order to develop some series approximations for the single and the product moments of order statistics from

an arbitrary continuous distribution.

4.2. David and Johnson's approximation

As has been mentioned earlier, the probability integral transformation u = F(x) transforms the order

statistic Xi:n from a continuous population with pdf f(x) and cdf F(x) into the uniform order statistic Ui:n.

Hence, by inverting the above transformation we have

-1 Xi:n = F (Ui:n) = G(Ui:n) (4.8)

which, upon expanding in a Taylor series about the point E(Ui:n) = Pi' yields

Xi:n = Gi + Gi (Ui:n -Pi) + i Gi (Ui:n -pi + ~ Gj' (Ui:n -pi 1 iv 4

+ 24 Gi (Ui:n -Pi) + ... ; (4.9)

G. denotes G(p.), G~ denotes ~ G(u) I _ ,etc. Now taking expectation on both sides of (4.9) and upon 1 1 1 u u-Pi

using the exact and explicit expressions for the central moments of uniform order statistics, we obtain

Piqi "Pi qi [1 m 1 iV] Il·. ~ G. + ~ G. + --2 'f(q.-p.) G. + tT p.q.G. l.n 1 ",\n+"'J 1 (n+2) :J 1 1 1 0 1 1 1

Pi qi [1 m 1 { 2 } iv + ~ - 'f (q.-p.)G. +.,. (q.-p.) - p.q. G. (n+2):J :J 1 1 1 't 1 1 1 1 1

+ ~ q.p.(q.-p.) G~ + ~ p? q? G~i]. (4.10) 011111 'tol11

Similarly, we may consider the series expansion for X;:n obtained from (4.9). Then by taking

expectation on both sides and then subtracting the expression for Iltn obtained from (4.10), we get

Var(X. ) = cr· . l:n l,en

Piqi (G,)2 Piqi [2( ) G' G" {G' Gm 1 (G")2}] ~ r.;-:;->r\ • + --2 q.-p. . . + p.q. . . + ~ . \n+"'J 1 (n+2) 1 1 1 1 1 1 1 1 ,t. 1

+ ~ [-2(q.-P.) G~G': + {(q.-p} - p.q.} (n+2):J 1 1 1 1 1 1 1 1

76

David and Johnson (1954) have worked out similar series approximations for the first four cumulants

and cross-cumulants of order statistics. Realize that they have given these approximations in inverse

powers of n + 2 (up to order 3) as they found it then convenient to combine terms of higher order. It is

also important to note that

G~ =~G(U)I = 1 I =~ 1 uU u=Pi f(l<l(u» u=Pi ~\ViJ

(4.13)

which is just the reciprocal of the probability density function of X evaluated at G .. This form of Gi in , 1

(4.13) allows us to write down the higher order derivatives of Gi without great difficulty in most cases. For

example, for the standard normal distribution, making use of the property that f' (x) = -x f(x) we have

1 G. 1+2G~ G' - G" - 1 G- 1

i -~' i - (f(G.)}2' i = (f(G.) }3' 1 1

G (7 602) 7 + 46 G~ + 24 G~ oiv _ i + i GV _ 1 1

i - {f(G.)}4' i - {f(G.)}5 1 1

. G.(127 + 326 G~ + 120 G~ GV1 _ 1 1 1 tc

i - (f(G.)}6 ,e . 1

For the case of the logistic distribution with

e-x 1 f(x) = 2 and F(x) = --, - < x < .. ,

(1+e -x) l+e-x (4.14)

however, Gi and its derivatives may be worked out rather easily. This is to be expected after all as the

logistic distribution admits ,an exact and explicit form for the inverse cumulative distribution function. For

example, we have in this case

77

As illustrated by David and Johnson (1954) and also by various other authors. this simple and straight

forward approximation procedure works well in most cases. However. the method does not usually provide

satisfactory results for extreme order statistics. In the case of extreme order statistics. the convergence of

this approximation to the true value may be very slow and even nonexistent in some cases. One may refer

to Section 8 for more details on this issue.

4.3. Clark and Williams' ftPlU"Oximation

Clark and Williams (1958) have developed some series approximations which are quite similar to

those of David and Johnson (1954). Approximations given by Clark and Williams (1958) make use of the

exact expressions of the central moments of uniform order statistics where the i'th central moment is of

order {(n+2)(n+3) ... (n+i)}-1; David and Johnson's (1954) approximations. on the other hand. are based on

the approximate expressions of the central moments of uniform order statistics which are in inverse powers

of n+ 2. as mentioned earlier in Section 2.

Thus. by starting with the series expansion of Xi:n given in (4.9) and then by using equation (4.13)

and the exact expressions of the central moments of uniform order statistics. we obtain

E(Xi:n) = Ili:n

e p.q. {3(e)2 - f.f'.'} p.q. (q. - p.) -0 1 1 1 + 1 11 11 1 1 - i - 2e \n+2J 3f. (n+2)(n+3)

1 1

{lOflifi - ff r; - 15(fi )3} npfqf + Piqi (2-5Piqi)

+ 8f? (n+2)(n+3)(n+4) + .... 1

(4.15)

where 0i = O(Pi)' fi = f(Oi)' fi = f' (Oi)' etc. Proceeding similarly. we also obtain

1 p.q. f~ 2p.q.(q. -p.) {15(f~)2 f'.' },np~q~ + p.q. (2-5p.q.) Var() 111 111 1 1 1, 11 11 11 Xi:n =? {ii+2)' - t. (n+2)(n+3) + ~ - e (n+2)(n+3)(n+4)

1 1 1 1

(e)2 p~q~ 1 1 1 -4£f (n+2)2 + ....

(4.16)

3 1 2p.q.(q.-p.) ge np~q~ + p.q.(2-5p.q.) EfV .... , ) _ 1 1 1 1 1 1 1 1 1 1 1

'''''i:n t"'i:n - 7 (n+2)(n+3) - if. (n+2)(n+3)(n+4) 1 1

and

3e p~q~ 1 1 1 +~~+ ... ,

2fj" (n+2)

78

(4.17)

4 3 np~q~ + p. q. (2-5p.q.) E(X ) - 1 1 1 1 1 1 + (4 IS)

i:n - ~i:n - 7. (n+2)(n+3)(n+4) .... . 1

From the above results we easily see the well known fact that if both i and n increase with i/n

remaining fixed, the asymptotic distribution of Xi:n has the mean and variance as Gi and Piqlntf, respect­

ively. Furthermore, from equations (4.16) - (4.1S) we also have

E(Xi:n - ~i:n)3 _ 1 {2(qi -Pi) 3JPiq~ fi} (Var(X. ) }3/2 - In ~p.q. - f. + ....

l:n 1 u 1 1 1

and

4 E(Xi :n - ~i :n) { 5n + 12} ----..... 2- = 3 1 - (n+3)(n+4) + .... (Var(Xi :n ) }

(4.19)

(4.20)

Realize that the known result that for large n the distribution of Xi:n is approximately normal is apparent

from equations (4.19) and (4.20).

Starting similarly with the series expansion of Xi:nXj:n and then using the exact expressions of the

central product moments of uniform order statistics, Clark and Williams 095S) have also derived approxi­

mations for the covariance and the correlation coefficient between the order statistics Xi:n and Xj:n as

C (X X ) - 1 ~iq~ fi Piqj (qi - Pi) L Piqj (qt - Pj) ov i:n' j:n - fi1j n+ - f£ (n+2)(n+3) - f. ~ (n+2) n+3)

1 J 1 J

+ {3(fP2 - fli } np~qiqj + Pi~i(2-5Piqi) 2f?f. (n+2)(n+3 (n+4)

1 J

{3(e)2 - f.f':} np.p.q~ + p.q.(2-5p.q.)

+ J JJ l~J Ii n 2f.f?n+2)(n+3 (n+4)

1 J ef~ np.q.(q. - q. + 3p.q.) + p.q.{1 + 5(p. + q.) - 15p.q.} +~ 1] 1 ] 1J It 1]. 1] 4?? (n+2)(n+3) n+4)

1 J

[e p.q. {3(e)2 - f':} p.q. (q. - p.) 1 11+ 1 1 111 1

- 2f~ ('ii+2)" 3f~ (n+2)(n+3) 1 1

{1Of.f~f': - i?-r. - 15(e)3} np~q~ + P.q.(2-5P.q.)] + III 11 1 11 11 11 Sf? (n+2)(n+3)(n+4)

1

[f~ p.q. {3(e)2 - f':} p.q. (q. - p.) -.-L:..L.L+ J J ~ J t J - 2? (n+2) 3f~n+2) n+3)

J J

{1Of.f~ f': . - f~f~' - 15(e)3} np~q~ + p. q . (2-5P.q.)] JJJ JJ J JJ J~ P + 7 (n+2)(n+)(n+4 + ...

Sf. J

(4.21)

79

and

[P.q.] 1/2 [ 1 {(e)2 2e f~

p(X. ,X. ) = ~ 1--- 1 p.q. -~.q. I:n J:n qiPj 4(n+2) rr- 1 1 "ffrfl J

(e)2 }] + TPjqj + ....

J

(4.22)

Formula in (4.22) has also been derived by David and Johnson (1954) from equations (4.11) and (4.12). A

comparison of the respective formulae in Sections 2 and 3 reveals that the approximations provided by

David ad Johnson (1954) are lot easier to use in practice.

4.4. Plackett's approximation

The methods of approximation discussed in Sections 2 and 3 have both been developed by applying

the probability integral transformation and then using the known moments of order statistics from the uni­

form distribution. Plackett (1958), instead, has proposed a method based on the logit transformation which

transforms a continuous order statistic Xi:n into the order statistic Li:n from the standard logistic distribu­

tion and then develops some approximations by using the 'explicit expressions of the moments of logistic

order statistics.

To illustrate the method of approximation due to Plackett (1958), let us denote Li:n (1 ~ i ~ n) for the

i'th order statistic in a sample of size n from a standard logistic distribution with pdf and cdf as given in

(4.14). Then the moment generating function of Li:n is given by

[ tLi:n] n! II [ u ] t i-I n-i E e = (I-I)! (n-I)! 0 r=u u (l-u) du

_ n! r(i+t) [,(n-i-t+l) - (x-l)!(n-I)! f(n+l)

= r(i+t) r(n-i-t+ 1)f['(i) ['(n-i+l). (4.23)

From (4.23), by taking logarithms, differentiating with respect to t and then setting t = 0, we obtain the

cumulants of Li:n as

K~) = ~ in r(i+t) + -i. in [,(n-i+l-t) I t-O I.n d~ dtJ -

= '¥G-l)(i) + (-l~ '¥G-l)(n_i+l), (4.24)

where ,¥(O)(z) = '¥(z) = ~ en r(z) = [" (z)f['(z) is the psi or digamma function, and '¥G-l\z)

j-l = ~ '¥(z) are the derivatives of the psi function which are referred to as polygamma functions. These

dzJ-functions have been tabulated quite extensively by Davis (1935) and Abramowitz and Stegun (1965). For i

> (n+l)/2, for example, we derive from (4.24) that

i-I

K~l) = \' 11k I:n 1. ' (4.25)

k=n-i+l

80

2 i-I n-i

Kf~~ = j- - l 1/k2 - l 1/k2, (4.26)

k=l k=l

i-I K~3) = 2 l:n (4.27)

k=n-i+1 and

4 i-I n-i K~4) = 21t _ 6 \' 1/k 4 - 6 \' 11k 4 (4.28) l:n D L L·

k=l k=l As has been mentioned eader, the logit transformation L = fn[F(x)/{l-F(x)}] transforms the order

statistic Xi:n from a continuous population with pdf f(x) and cdf F(x) into the logistic order statistic Li:n.

Then by considering Xi:n as a function of Li:n and expanding in a Taylor series about the point E(Li:n) =

K~l) we have l:n'

X. = x(O) + x(l) [L. _ KP)] + 1 x(2) [L. _ K~l)] 2 l:n l:n l:n! l:n l:n

o l:n l:n 24 l:n l:n ... , (4.29) + 1 x(3) [L. _ KP)]3 + 1 x(4) [L. _ K~1)]4 +

where xU) is the value of the j'th derivative of x with respect to L at L = Kf~~. Now taking expectation on

both sides of (4.29) and upon using the exact and explicit expressions for the cumulants of logistic order

statistics given in (4.24) - (4.28), we obtain

J.l.. ~ x(O) + 1 x(2) K~2) + 1 X(3)K~3) + 1 x(4){K~4) + 3 [K~2)) 2}. l:n ! 1: n 0 1: n 24 1: n 1: n (4.30)

The coefficients in the above approximation are easy to obtain as in the case of David and Johnson's (1954)

approximation. For example, for the standard normal distribution with pdf f(x) and cdf F(x), we have

K(l) (1) i: n

x(O) = F-1 {e Ki :i(1+e )},

x(l) = F(1-F)/f,

x(2) = x(l){xx(1) - (2F-1)},

x(3) = (x(l))3 + 2xx(l)x(2) + x(2)(1-2F) - 2x(1)P(1-F)

and

x(4) = 5(x(1))2 x(2) + x(3){2xx(l) - (2F-1)} + 2x(2){xx(2) - 2F(1-F)} + 2x(1) (2F-l)F(1-F);

these derivatives are all bounded. Suppose we include the first j-l terms in the series expansion for J.l.i:n

given in equation (4.30), then the absolute value of the remainder after j-l terms is at most ~ max IxCD I

E I Li:n - K? ~ Ij· Since E I Li:n - K? ~ 12j is known and also that 1 1

{ElL. -KP) 12j- 1}2J=r :=;; {ElL. _K~l) 12j}2f l:n 1: n l:n 1: n '

we realize that we will be able to present bounds to J.l.i:n for all values of j.

81

4.5. Saw's error analysis

We shall consider here David and Johnson's (1954) series approximation for the mean of normal order

statistic discussed in Section 2 and illustrate Saw's (1960) method of deriving bounds for the error which

results in using only a finite number of terms in the series.

From (4.8), we may write

2m [ ] \' 1 (t) t

~i:n = E Xi:n = l 'iT 0i E(Ui:n -Pi) + ~m' t=O

(4.31)

t where p. = E(U .. ) = iI(n+l), O(u) = p-l(u), O. = O(p.) and dt) = ~t O(u) I ; R2 ,the remainder

I I.n I I I du u=Pi m after 2m terms, is the difference between the true value of ~i:n and the sum of the series to 2m terms. Let

us consider the Taylor series expansion for the function x = p-l(u) = G(u) about the point u = Pi given by

2m x = O(u) = \' !.. dt) (u-p.)t + R! . l t! I I --2m (4.32)

t=O Using the Darboux form for the remainder R2m in (4.32), we may write

1 R! = 1 I (1-{)2m d 2m+1)(p.+ ~(u-p.» d~. --2m "{2iii}T 0 I I

(4.33)

Now multiplying both sides of (4.32) by the function ui- 1(1-u)n-i/B(i,n-i+l), integrating over u in the

interval (0,1), and comparing the resulting equation with (4.31), we get

1-" n-l, Jl 1 . 1 . R2m = 0 B(I,n-l+I) u (1-u) R2m(u) du;

combining this with equation (4.33), we have

1 II II "1' 2m (2m+l)[ -,-1' ] 1 i-I n-i l' ~m ="{2iii}T 0 0 (1..,) G Pi .... (u-Pi) B(I,n-l+I) u (1-u) du d .... (4.34)

As noted in Section 2, we see that for j ~ 1

~ G) . ~ x(u) = G (u) = H.(x)/(f(x)Y, dJ J where Hix) is a polynomial of degree j-l in x which is an even or odd function depending on whether j is

odd or even. We have, for example,

2 HI (x) = 1, ~(x) = x, ~(x) = 1 +2x ,

H4(x) = 7x + 6x3, H5(x) = 7 + 46x2 + 24x4,

H6(x) = 127x + 326x3 + 120x5,

etc., and it is easily verified that these polynomials satisfy the recurrence relation

Hj+l(x) = j x Hix) + Hj(x), (4.35)

where Hj (x) = ~ Hj(x). We also note that

GG)(u) = (_1~-1 GG)(1-u). (4.36) ... . G) ......

Denoting now G. (u) for J G (u), we see immediately that G. (0) = 0 and GJ. (u) ... 00 as u ... 1. J J

82

Moreover, for i < u < I, we note that O;m+ 1 (u) = u2m+ 1 O<2m+ I)(u) is the product of two increasing

functions and. hence, is an increasing function of u. Next, for 0 < u < i (that is, x < 0), we consider

lk O;m+l(u) = (2m+l) u2m O<2m+I)(u) + u2m+1 O<2m+2)(u)

H_ (x) H_ (x) = (2m+l) u2m --.lm+l + u2m+1 --.lm+2 (f (x»Zm+l (f (x) )Zm+Z

u2m {H2m+ 1 (x) } = (f( x»Zm+Z ~m+2(x) (2m+l) f(x) H2m+2(x) + u

= Kl (u) ~(u), (4.37)

where

u2m H2m+ 1 (x) K1(u) = (f(x» Zm+Z H2m+2(x) and ~(u) = (2m+l)f(x) H 2m+2(x) + u.

Now for 0 < u < i, we have x < 0 and, consequently, Kl (u) < O. Noting that ~(O) = 0, we consider

d ~(u) f (x) [ 2 dx 2" - Him+2(x) - (2m + I) H2m+l(x) ~m+2(x)

Him+2(x)

+ (2m + I) x ~m+l(x) ~m+2(x) + (2m + I) ~m+l(x) H2m+2(X)] 2 = -f(x) A2m+1(x)lH2m+2(x) (4.38)

upon using the relation in (4.35), where

A2m+1(x) = (2m+l) ~m+l(x) H2m+2(x) - (2m + 2) H2m+l(x) ~m+2(x) or, equivalently,

2 A2m+l(x) = (2m+l) ~m+l(x) ~m+3(x) - (2m+2) Him+2(x).

From (4.39), for example, we get

A1(x) = I,

A3(x) = 21-16 x2 + 12 x4,

A5(x) = 4445 - 6664 x2 + 8076 x4 + 1344 x6 + 2880 x8,

~(x) = 3,884,041 - 8,666,072 x2 + 14,468,040 x 4

6 8 10 12 + 6,808,896 x + 24,184,656 x + 11,093,760 x + 3,628,800 x ,

2 4 A9(x) = 9,580,522,329 - 28,374,712,624 x + 60,246,981,384 x

+ 54,671,037,120 x6 + 292,158,113,616 x8 + 337,984,717,824 xlO . 12 14 16

+ 259,516,161,792 x + 88,044,104,640 x + 14,631,321,600 x .

(4.39)

We note that A2m+l(x) is positive defmite in all these cases since the first three terms (that is, the terms

involving xO, x2 and x 4) form a positive definite quadratic in x2. Based on (4.37) and (4.38), therefore,

* we conclude that 02m+ 1 (u) is an increasing function in u in the interval (0,1) (at least for the cases m = 0,1,2,3,4).

For the standard normal distribution, since Ili:n = -iln-i+l:n let us consider the cases when i

> (n+I)/2; that is, Pi > 1/2. Making a substitution v = Pi + ~(u-Pi) in equation (4.34), we get

83

_ 1 [I Pi IV 2m (2m+l) 1 i-I n-i ~m - {2m}T - 0 0 (u-v) G (v) B(I,n-I+I) u (l-u) du dv

+ I Pi II ( - )2m G(2m+l)( ) 1 i-I (I )n-i d d] o v u v v B(I,n-1+I) u -u u v.

Replacing v by I-v, u by l-u and then using the symmetrical property of d 2m+l)(v) given in (4.36) in the

second integral, we obtain

R - 1 If ( )2m G(2m+l)( ) 1 {n-i(1 )i-l _ i-l(1 )n-i} d d 2m - {2m}T v-u v B(I,n-l+I) u -u u -u v u o<ll<v<qi

1 ·IPi IV ( )2m G(2m+l)( ) 1 i-l(1 )n-i d d - {2m}T q. 0 v-u v B(I,n-I+I) u -u u V

1

= 11 - 12 (say); (4.40)

note that 11 and 12 are both positive since the integrands are everywhere positive. Now by considering

1 II u 2m 1 * 1 {n-i i-I i-I n-i} 11 = {2m}T (1 - y) y G2m+ 1 (v) B(I,n-I+I) u (I-u) - u (l-u) dv du o<u<v<qi

* and using the property that G2m+1(v) is an increasing function in v, we have

II u 2m 1 1 { n-i i-I i-I n-i} (1 - y) y B(I,n-I+I) u (l-u) - u (l-u) dv duo

Since

r.. 2 d r.. 2 q. (q. _u)2m+l 1(1 _~) m ~ < 1(1_~) m -4 dv = ---=1_---..=-_,

U V V U v v (2m+l) qIm u'

we have

( 2m+l) qi G i r'li 2m+1 1 1 { n-i i-I i-I n-i}

11 < (2 m+ 1 )! J 0 (qi - u) ii B(I,n 1+1) u (I-u) - u (l-u) duo (4.41)

By considering 11 once again from (4.40) and using the property that d 2m+I)(v) is a decreasing function

of v in (O,qi < i), we have

O<2m+l) i II m+l 1 {n-i i-I i-I n-i} 11 > (2m + I)! (v-u) B(I,n-l+l) u (I-u) - u (l-u) du dv

o<u<v<qi which, when performed the integration over v, yields

G(2m+l) i r'li 2m+l 1 { n-i i-I i-I. n-i}

11 > (2m+1)! JO (qi -u) B(I,n-I+I) u (l-u) -u (l-u) duo (4.42)

By considering 12 from (4.40) and using the property that d 2m+l)(v) is an increasing function of i, we have

(2m+l) Gi JPi IV 2m 1 i-I n-i

12 < (2m)! q. 0 (v-u) B(I,n-I+I) u (1-u) du dv 1

and

84

(2m+ 1) 1 G (,;) JPi IV 2m 1 i-I n-i

J2 > . (2m)! 0 (v-u) B(I,n-l+l) u (l-u) du dv. q.

I

Writing both the double integrals as

ti r du dv == ti dv 1i du + ti ti dv du qi 0 qi 0 qi u

and integrating out over v, we obtain after simplification

_ G (2m+l) J < i [E(U _ )2m+l + r<Ji ( _ )2m+l 1 2 (2m+1 )! i:n Pi JO u qi B"""(I-,n=--'17+1")

and

d2m+l)(I) J > - 2" [E(U _ )2m+1 + r<Ji ( )2m+l 1 2 (2m+I)! i:n Pi J 0 u-qi B(l,n-l+I)

{un- i (1-u)i-l_ ui- 1(l-U)n-i} dU]'

Combining equations (4.42) and (4.43), we obtain from (4.40) that

d 2m+l) R > i E(U _ )2m+1

2m (2m+ I)! i:n Pi .

Similarly, combining equations (4.41) and (4.44) we derive an upper bound for R2m from (4.40) as

q.G ~ 2m+l)

R2m < b~+ I )! {V 2m+l(i,n-i+l;qi) - V 2m+l(n-i+l,i;qi)}

d 2m+ 1)(1)

- (2 m+l) T {W 2m+l(i,n-i+l;qi) - W 2m+l(n-i+l,i;qi) -E(Ui:n - p/m+l}, where

Wc(a,b;v) = I: (u_v)c ~ ua- 1 (l_u)b-l du

and

Iv 1 c 1 a-I b-l V c(a,b;v) = 0 u (u-v) B'(ii';D)' u (1-u) duo

For computational ease, we give the following recurrence relations satisfied by the above quantities

involved in computing the bounds derived in (4.45) and (4.46):

n+c+2 E(U. _ .)c+l = n-2i+l E(U. _ .)c _ i(n-i+l) E(U. .)c-l c I:n PI ----n+T'" I:n PI (n+I)2 I:n -PI '

V c+ 1 (a,b;v) + V V /a,b;v) = W c(a,b;v)

and

(a+b+c) Wc+1(a,b;v) = a - v(a+b) + c(I-2v) Wc(a,b;v) + cv(1-v) Wc_1(a,b;v),

where we take

1 a-I b-l c W l(a,b;v) = - 'D'fnh\ V (I-v) c- u,a, VI if c = 0

= Iv(a,b) if c = I,

and

(4.43)

(4.44)

(4.45)

(4.46)

85

a + b - 1 VO(a,b;v) = a-I -\,(a-l,b);

-\,(a,b) is Karl Pearson's incomplete beta function defined by

Iv 1 a-I b-l -\,(a,b) = 0 ma,l» u (l-u) duo

From (4.45) and (4.46), therefore, we will be able to get the bounds for the error involved in approxi·

mating Ili:n by David and Johnson series up to 2m terms. Saw (1960) computed the bounds for the error

involved in David and Johnson series as well as Plackett series. Upon comparing them, he observed that

term for term the Plackett series converges a little more rapidly than does the David and Johnson series.

However, as rightly pointed out by Saw (1960), the slight superiority of the Plackett series in respect of

convergence is quite overshadowed by the computational advantages of the David and Johnson technique.

4.6. Su&iura's ortho&onal inverse expansion

Sugiura (1962) applied an orthogonal inverse expansion approach to obtain approximations as well as

bounds for the moments Ilf~~ (1 :s; i :s; n) for an arbitrary distribution. As explained in Section 3.1, by

denoting (q,.(u)} ":'=0 for a sequence of complete orthonormal functions in L 2(0,1), we have for an arbitrary J J-

standardized distribution with Jl = 0 and cr = 1 (see Theorem 3.1)

k {k } 112 { k } 112 Ill .. - ~ a.b·l:s; 1 - ~ a~ B(2i-l,2n-2i+l) -1 - ~ b~ ,

l.n £ J J £ J [B(·· I)] 2 £ J j=1 j=1 l,n-l + j=1

(4.47)

where

Jl 1 i-I n-i bj = 0 B(I,n-1+I) u (1-u) <l>j(u) duo

If the parent distribution is symmetic about zero, we may obtain improved bounds for Ili:n as (see

equation (3.27»

k

Illi:n - L ~j+lb2j+ll j=O

:s; {I _ ~ 2. }1!2 { B(2i-l,2n-2i+l) - B(n,n) _ ~ b2. }1!2. .=0£ ~J+l 2[B(i,n-i+l)] 2 .=0£ 2J+l J- J

(4.48)

Further, if the parent distribution has a finite fourth moment, we may proceed similarly and obtain bounds

for 1I~2) as '-1 :n

86

k

I (2) ~ * I ~i:n - L. ~jb2j j=O

k k

~ {E(0) - 1: a;~}1!2 {B(2i-l'2n~2i+~) + ~(n,n) - 1: b~j}l/2, j=O 2[B(l,n-1+l)] j=O

where

* Jl -1 2 a· = (F (u)) cp.(u) duo J 0 J

If we take the Legendre polynomials on (O,l) defined by

j = 0,1,2, ...

j

= {2j+T 1: (-l~-£ ({) (j;£) u£

£=0 for the complete orthonormal system, then we have

aj = {2j+T i (-l~-£ (~ (j;£) r F-1(u) / du £=0 0

j

= {2j+T 1: (-l~-£ ({) (j;£) ~£+l:£+l/(f+l), £=0

a; = {2j+T i (-l~-£ (~ (j;£) f (F-l (u)}2 / du

£=0 0

and

j

= {2j+T 1: (-l~-£ (~ (j;£) ~~~L£+l/(f+l), £=0

b. = ~ ~ (-l~-£ (j~ (j+£) 1 Jl £+i-l(l_ )n-i d J 1 "'J ' , L e £ B(l,n-HI) 0 u u u

£=0

j

= f2J+1 1: (-l~-£ (~ (j;£) B(f+i,n-i+l)/B(i,n-i+l)

£=0

j

= {2j+T 1: (-l~-£ ({) (j;£) (~i~)~~~it~f~!b· £=0

*

(4.49)

(4.50)

(4.51)

(4.52)

Realize that the Fourier coefficients aj and aj involved in deriving the bounds and approximations in

(4.48) and (4.49) may be computed from equations (4.50) and (4.51) by using the first and the second

moments of extreme order statistics in small sample sizes.

For the standard normal distribution, for example, we have

II -1 1 F (u) u du = ~2'2!2 = -, o . 2[7t

87

II 1 2 1 p- (u) u du = ~3'3/2 = -, o . 2[1t

II 1 3 1-1 p- (u) u du = ~4.J4 = -- Cos (-1/3),

o . 41t[1t

II 1 4 1 { 1 -1 } p- (u) u du = ~5'5/5 = - 1 -it Cos (-113) o . 2[1t and

1 fo P-l (u) uS du = ~6:d6 = 1.267206361/6;

the value of ~6:6 was taken from Teichroew (1956). Making use of the above results in (4.50), Sugiura

(1962) computed the bounds and approximations for ~i:n from (4.48) for n = 10 and 20. The results for the

cases n = 10 and n = 20 are presented in Tables 4.1 and 4.2, respectively. Prom these tables, we see that

the approximations are uniformly good for all i; this is not surprising since the orthogonal expansion gives

the best approximation in the mean.

10

9

8

7

6

20

19

18

17

16

15

14

13

12

11

*

Table 4.1. Approximations and bounds for ~i: 10 in the

k = 0

1.38 ± 0.17

1.08 ± 0.09

0.77 ± 0.14

0.46 ± 0.13

0.15 ± 0.06

Table 4.2.

k = 0

1.53 ± 0.35

1.37 ± 0.17

1.21 ± 0.15

1.05 ± 0.16

0.89 ± 0.18

0.73 ± 0.20

0.56 ± 0.21

0.40 ± 0.19

0.24 ± 0.14

0.081 ± 0.051

* standard normal distribution

k = 1 k = 2

1.527 ± oms 1.5384 ± 0.0005

1.030 ± 0.035 1.0032 ± 0.0026

0.651 ± 0.008 0.6527 ± 0.0048

0.357 ± 0.028 0.3775 ± 0.0024

0.113 ± 0.016 0.1246 ± 0.0032

Approximations and bounds for ~i:20 in the

* standard normal distribution

k = 1 k = 2

1.796 ± 0.082 1.856 ± 0.016

1.468 ± 0.067 1.433 ± 0.032

1.186 ± 0.075 1.131 ± 0.012

0.945 ± 0.057 0.907 ± 0.018

0.739 ± 0.034 0.734 ± 0.022

0.564 ± 0.037 0.587 ± 0.013

0.413 ± 0.056 0.454 ± 0.008

0.281 ± 0.065 0.325 ± 0.019

0.163 ± 0.053 0.197 ± 0.020

0.054 ± 0.021 0.066 ± 0.008

Tables are reproduced with the permission of the author.

Exact value

1.53875

1.00136

0.65606

0.37576

0.12267

Exact value

1.867

1.408

1.131

0.921

0.745

0.590

0.448

0.315

0.187

0.062

88

Sugiura (1964) has extended this approach to derive bounds and approximations for the product

moment J.1ij:n; one may also refer to Mathai (1975, 1976) for some further generalizations.

4.7. Joshi's modified bounds and apJ!fOximations

The orthogonal inverse expansion method of Sugiura (1962) discussed in the previous section requires

that the population distribution has a finite variance. By a simple generalization of this method, Joshi

(1969) has shown that, even with less stringent conditions, similar bounds and approximations may be

derived for all finite moments of order statistics.

Theorem 4.1: Let '0 == 1, '1' '2'''' be an orthonormal system in L 2(0,1) and let J.1i;ll:2P+2q+ 1 be finite for some integral p,q ~ O. Then for 1 S; i S; n

k

IB(R+i,Q+n-i+l) J.1p+i:p+q+n - l aJ~bJ'1 (l,n-HI) '=0

S; {B(2P+l,2q+l) tJ2) J~ ~ a~2}tn. {B(2i-l,2n-2i+l) _ ~ b~}I!2, r-:lp+l:2p+2q+1 j~ J [B(i,n-i+l)]2 j~ J

(4.53)

where

1 aj = 10 p-l(u) uP(l-u)q 'lu) du

and bj is as defined in (4.52).

Proof. The result follows immediately upon applying (3.21) with

-1 P q 1 i-I n-i f(u) = P (u) u (1-u) and g(u) = B(l,n-t+l) u (1-u) .

Remark: Theorem 4.1 shows that the bounds and approximations for J.1p+i:p+Q+n (I S; i S; n) may be

b . ed 'ded th (2) . fini o tam proVl e moment J.12p+l:2p+2q+IIS teo In tenns of the pdf f(x), a sufficient condition for

this is (Sen, 1959)

(lx I2/(2P+I) f(x) dx < 00

forq~p~O.

If the parent distribution is symmetric about zero, we may obtain some improved bounds and

approximations as given in the following theorem.

Theorem 4.2: Let the parent distribution be symmetric about zero. Let '0 == 1"1"2"" be a complete

orthonormal system in L2(0,1) such that 'j(u) = (-l~ 'P-u), j = 1,2, .... If J.1i;ll:4m+l is finite for some

integral m ~ 0, then for 1 S; i S; n

k

I B(m+i,m+n-i+l) \', I B (l,n-HI) J.1m+i:2m+n - L. Il2j+l b2j+l

j=O

k

~ {B(2m+l,2m+l) ~i;ll:4m+l -.l a2I+l}1I2 J=O k

{B(2i-l, 2n-2i+l) - B(n,n) _ \' b2. } 112, 2[B( i,n-i +1)]2 j~ 2J+l

where aj is as in Theorem 4.1, with p = q = m.

89

Moreover, if ~i!ll:4m+ 1 is finite for some integral m ~ 0, then for 1 ~ i ~ n k

I B(m+i,m+n-i+l) ( 2) \'" I B (l,n-HI) ~m+i:2m+n - l a2j b2j

j=O

k

S {B(2m+l.2m+1) .l!lum+l- j~ '27]"2 k

{B(2i-l,2n-2i+l) + B(n,n) _ l b2.} 112, 2[B(i,n-i+1)]2 '=0 2J

J-where

"II 12m m aj = 0 {r (u)} u (l-u) «Plu) duo

ftQQf. To prove (4.54), we take

-1 m m 1 i-I n-i f(u) = F (u) u (l-u) and g(u) = B(I,n-I+I) u (1-u) ,

apply (3.23) with j = {1,3, ... ,2k+l,0,2,4,6, ... } and use the results that

1 a2j = Io F-1(u) um(l_u)m «P2lu) du = 0

and

.. 2 II 1 { }2 l b2j = 4" g(u) + g(l-u) du . 0 0 J=

_ B(2i-l , 2n-2i+ I) + B(n,n) - 2 [B(i,n-i +1)] 2

Similarly, to prove (4.55), we take

( -12m m 1 i-I n-i f(u) = F (u)} u (l-u) and g(u) = B(I,n-l+I) u (l-u) ,

apply (3.23) with J = {0,2,4, ... ,2k,I,3,5,7, ... } and use the results that

1 a2j+1 = Io (F-1(u)}2 um(l-u)m «P2j+l(u) du = 0

and .. 11 1 2 = 04" (g(u) - g(l-u)} du

_ B(2i-l,2n-2i+l) - B(n,n)

- 2[B(i,n-i+l)] 2

(4.54)

(4.55)

If we now take the sequence of Legendre polynomials on (0,1) for the complete orthonormal system

90

{CPlu)}j=o' then we have

aj =.{2f+1 i (_l~-l (}) (j;l) f F-1(u) ul+m(I-u)m du

l=O 0

j

= ro+l 1: (_l~-l (~ (j;l) B(l+m+l,m+l) Jll+m+l:l+2m+l'

l=O

aj = ro+l i (-1~-l (~ (j'fl) f {F-1(u)}2 ul+m(I-u)m du l=O 0

j

(4.56)

= ro+l 1: (_l~-l (~ (j'fl) B(l+m+l,m+l) Jl~~~+I:l+2m+l' (4.57)

l=O and bj is as in equation (4.52).

For the Cauchy distribution with pdf

1 f(x) = ~, - DO < X < .. , '/t(l+x )

Barnett (1966) has shown that Jli~!') is finite for all i' < i :S; n-i'. In particular, Jl? § is finite and hence

(4.54) is applicable with m = 1. We may thus get bounds for Jli+l:n+2' 1 :S; i :S; n, which are all the finite

moments in this case. For the case m = I, we note from (4.56) that aj is a linear function of the moments

Jll+l:l+2 in small samples which are usually available; see, for example, Tietjen et a1. (1977) for the

normal distribution, and Barnett (1966) and Rider (1960) for the Cauchy distribution. Making use of

Theorem 4.2, Joshi (1969) computed the bounds and approximations for Ili:n (2:S;i:S;n-l) when n = 10 and 20

for the standard normal and Cauchy distributions. These are presented in Tables 4.3 - 4.5. From these

tables, we see that the approximations and bounds are remarkably good for all i (2 :S; i :S; n-l), especially

for the Cauchy distribution.

As mentioned earlier, we can obtain several different sequences of bounds all converging to

lli+m:n+2m by choosing different values of m and n in (4.54). Unfortunately, there is no theoretical way

of determining the choice of m, for which the difference between the true value of Jli+m:n+2m and its

approximate value is minimum for a fixed value of k. After performing some numerical computations for

the normal distribution with n + 2m = 20 and 50, Joshi (1969) observed that the case when m = 1 gives "

better results than the cases when m = 0 or m = 2 for the second approximation (that is, when k = 1).

Table 4.3. Approximations and bounds for Ili: 10 in the

* standard normal distribution

k=O k = 1 Exact value

9 1.2994 ± 0.2982 1.0041 ± 0.0028 1.00136

8 0.5304 ± 0.1325 0.6509 ± 0.0053 0.65606

7 0.2475 ± 0 1355 0.3788 ± 0.0031 0.37576

6 0.0743 ± 0.0567 0.1249 ± 0.0025 0.12267

9

8

7

6

91

Table 4.4. Approximations and bounds for lli:lO in the

k = 0

3.0529 ± 0.0727

1.2461 ± 0.0323

0.5815 ± 0.0330

0.1745 ± 0.0138

* Cauchy distribution

k = 1

2.9822 ± 0.0015

1.2749 ± 0.0028

0.6129 ± 0.0016

0.1866 ± 0.0013

Exact value

2.9814

1.2755

0.6132

0.1866

Table 4.5. Approximations and bounds for lli:20 in the

* standard normal and Cauchy distributions

Normal distribution Cauchy distribution

k = 1 Exact value k = 1 Exact value

19 1.4693 ± 0.0619 1.40760 6.2705 ± 0.0324 6.2648

18 1.1023 ± 0.0310 1.13095 3.0287 ± 0.0162 3.0293

17 0.9002 ± 0.0225 0.92098 1.9128 ± 0.0118 1.9140

16 0.7390 ± 0.0116 0.74538 1.3259 ± 0.0060 1.3268

15 0.5940 ± 0.0063 0.59030 0.9480 ± 0.0033 0.9484

14 0.4570 ± 0.0094 0.44833 0.6718 ± 0.0049 0.6720

13 0.3241 ± 0.0115 0.31493 0.4506 ± 0.0060 0.4506

12 0.1937 ± 0.0095 0.18696 0.2600 ± 0.0050 0.2599

11 0.0644 ± 0.0037 0.06200 0.0851 ± 0.0019 0.0850

*Tables are reproduced with the permission of the author

4.8. Joshi and Balakrishnan's improved bounds for extremes

By noting that the bounds for extreme order statistics obtained by Sugiura's method may not be sharp

and that the moments are usually tabulated for some typical values of n in case of large sample sizes, Joshi

and Balakrishnan (1983) and Balakrishnan and Joshi (1985) have developed a method of obtaining improved

bounds and approximations for moments of extreme order statistics by making use of few neighbouring

tabulated values. These bounds are of similar type as those of Sugiura (1962) discussed in Section 6.

To fix the ideas, let II be the moment for which bounds are required and m and p be any two ""n:n integers in the neighbourhood of n for which Ilm:m and IIp:p are tabulated. We can then write

92

_ II -1 (n-l m-l p-l ~. - c ~m.m - d ~p.p - F (u) nu - c mu - d pu 1 du, ~n . . 0

where c and d are constants so chosen that the error bound by applying the inequality in (3.21) for a (k+l) -

tenn approximation is as small as possible.

Then, upon applying (3.21) with

f(u) = F-1(u) and g(u) = nun- 1 - c mum- 1 - d puP-l

and the sequence of Legendre polynomials as the orthonormal system ("'k(u) lk=O' we obtain

k

I ~n:n - c ~m:m - d ~p:p - l aj bj I j=O

{ k} In. [2 2 2 2 2 k ] In. ~ E(X2) _ ~ a~ n + c m + ctL _ 2 cmn _ 2dpn 1 + 2 cdm1- ~ b~ , L J 2n=f 2m-l Tp=f m+n-l p+n- m+p- L J

j=O j=O where

II 1 aj = 0 F- (u) ",/u) du

and

II {n-l m-l -I} bj = 0 nu - c mu - d puP ",/u) du

= b. - c b. - d b. . J,n J,m J,P'

we have

j = N+T l (_l~-t (~2 ut(l_u~-t

t=O by applying Leibniz rule and, hence,

II s-1 b. = su ",.(u) du J,S 0 J

= N+T i (_I~-t (~2 t sus+t-l(l-u~-t du t=O 0 j

= N+T l (-1~-t (~ (S+~-I)/(s;j) t=O

(4.58)

(4.59)

(4.60)

(4.61)

- Mf+1 (s-I)(s-2) ... (s-j) (462) - 1"'J ~ ~ (s+I)(s+2) ... (s+J)" .

Equation (4.62) follows immediately by considering the identity (l-t~(l-t)-s = (1-t~-S and equating the

coefficients of J on both sides. Noting now that only the second factor on the RHS of the inequality in

(4.58) contains c and d and substituting for bj from equation (4.60) and minimizing the resulting expression

with respect to c and d, we obtain the optimal values of c and d for (k+l) - tenn approximation as

c = (R R - R R )/(R 2 - R R ) opt n,p m,p n,m p,p m,p m,m p,p and

93

d = (R R - R R )/(R 2 - R R ), opt n,m m,p n,p m,m m,p m,m p,p where

k R = ~ b. b. _ ab .

a,b l J,a J,b a + b - 1 j=O

Making use of these optimal values for c and d in (4.58), we may get an approximate value for Iln:n as

k c til + d til + ~ a.b. along with a bound given by the RHS of (4.58). op m:m op p:p j=O J J

If the parent distribution is symmetric about zero, we may obtain some improved bounds and approx­

imations as given in the following theorem.

Theorem 4.3: For positive integers a and b, let

H(a,b) = ~ {a + ~ _ 1 - B(a,b)}, where B(a,b) is the complete beta function. Then for symmetric distributions with mean 0 and variance 1,

we have for k ~ 0

k

Illn:n - c Ilm:m - d IIp:p - L ~j+ 1 b2j+ 11 j=O

k

< ([ - j~ ~j+ [fa (H(n,n) +02 H(m,m) + .2 H(p.p) - 2c H(m,n)

k }1!2 - 2d H(p,n) + 2cd H(m,p) - L b~j+1 '

j=O (4.63)

where aj and bj are as given in (4.59) and (4.60), respectively, and the optimal values of c and d are given

by

c = (S S - S S )/(S 2 - S S ) opt n,p m,p n,m p,p m,p m,m p,p and

with

k

Sa,b = L b2j+1,a b2j+l,b - H(a,b). j=O

Proof. To prove (4.63), we take

f(u) = F-1(u) and g(u) = nun- 1 - c mum- 1 - d puP-l,

apply (3.23) with J = (1,3, ... ,2k+l,0,2,4,6, ... ), and use the results that

J1 -1 ~j = 0 F (u) <l>2/u) du = 0

and

~ 1 2 L b~j = J {g(U) i g(l-U)} du '0 0 J=

94

= O(n,n) + c2 O(m,m) + d2 O(P,p) - 2c O(m,n) - 2d O(p,n) + 2cd G(m,p),

where G(a,b), for positive integers a and b, is

G(a,b) = ~ {a + ~ _ 1 + B(a,b)}. (4.64) Proceeding similarly, we can prove the following theorem which yields bounds and approximations

for the second moment of Xn:n.

Theorem 4.4: For positive integers a and b, let G(a,b) be as defined in (4.64). Then for distributions

symmetric about zero with variance I and finite E(X\ we have for k ~ 0

k

I~~~~ -c ~~~! -d ~~~~ - l a;jb2j I j=O

< (EO<"J -it ~) 1/2 (o(n .. ) + c2 O(m.m) + d2 G(p,p) - 2c O(m,n)

k } 112 - 2d G(p,n) + 2cd G(m,p) - l b~j ,

where, as before,

a: = Jl {F-1(u)}2 ,.(u) du J 0 J

and the optimal values of c and d are given by

= (T T - T T )/(T2 - T T ) Copt n,p m,p n,m p,p m,p m,m p,p and

d = (T T - T T )/(T2 - T T ) opt n,m m,p n,p m,m m,p m,m p,p' with

k

Ta,b = l b2j,a b2j,b - G(a,b). j=O

j=O (4.65)

By making use of the expression of the Legendre polynomials given in equation (4.61), Joshi and

Balakrishnan (1983) obtained an expression for the Fourier coefficients aj as

j

aj = ~ l (_I~-l ({) ~l+l:j+1' j ~ O. (4.66) l=O

Equation (4.66) expresses aj as a linear combination of ~I:j+l' ~2:j+I"'" ~j+l:j+1 with coefficients which are

* not too large unlike the expression given in (4.50). Similarly, we may write the Fourier coefficients aj as

j

a~ = ~ l (_I~-l ({) ~~~L+l' j ~ O. (4.67) l=O

By making use of the means of standard nonnal order statistics tabulated by Yamauti (1972) and the table

of second moments prepared by Teichroew (1956), Joshi and Balakrishnan (1983) have computed the coeffi-

* cients aj and aj for the standard normal case; these values are presented in Tables 4.6 and 4.7, respectively.

For large values of n, m and p, the beta tenn in the function H(a,b) in (4.63) becomes negligible and

95

hence can be ignored. This reduces (4.63) to a much simpler form. Then, Joshi and Balakrishnan (1983)

applied this simplified form of Theorem 4.3 to compute approximation and bound for Iln:n for n = 100

(100) 1000 by taking m = 400, P = 1000 and k = 13, and these are presented in Table 4.8. For

Table 4.6. Fourier coefficients aj for the standard

normal distribution

aj

1 0.9772050237

3 0.1830082402

5 0.0816989763

7 0.0477293675

9 0.0318804432

11 0.0230790883

13 0.0176307593

15 0.0139963491

17 0.0114376017

19 0.0095600272

21 0.0081467944

23 0.0070483945

25 0.0064146017

27 0.0059105768

* Table 4.7. Fourier coefficients aj for the standard

normal distribution

* aj

0 1.0000000000

2 1.2328088881

4 0.5211245856

6 0.3045144707

8 0.2055889844

10 0.1507706975

12 0.1166778189

14 0.0937736622

16 0.0775095317

18 0.0654684404

comparison purposes, we also present in this table the approximation and bound obtained by Sugiura's

method with k = 13 and the tabled values taken from the tables of Harter (1961) and Tippett (1925). From

this table; we see that the bounds and approximations obtained from Theorem 4.3 are considerably better

than those of Sugiura.

96

For large values of n, m and p, the beta tenn in the function G(a,b) in (4.65) becomes negligible and

hence can be ignored. By using this simplified form of Theorem 4.4, Balakrishnan (1980) computed the

approximation and bound for Il~~~ for n = 75 (5) 120 by taking m = 80, p = 120 and k = 9, and these are

presented in Table 4.9. Once again, for the purpose of comparison we present in this table the approxima­

tion and bound computed by Sugiura's method with k = 9 and the tabled values taken from Borenius (1966).

From this table, we see once again that the bounds and approximations obtained from Theorem 4.4 to be

better than those of Sugiura (1962).

Balakrishnan and Joshi (1985) have applied successfully Joshi's method to extend the results presented

in this section in order to derive improved bounds and approximations for the moments of the second

largest order statistic. It should be remarked here that the Fourier coefficients aj given in Table 4.6 can be

n used to approximate the sum of squares of normal scores, viz., S = l; Il~. ; see, for example, Ruben (1954),

i=l l.n Saw and Chow (1966), and Balakrishnan (1984).

Table 4.8. Approximations and bounds for Iln:n in the

standard nonnal distribution

n Sug!ura's bound Bound from (4.63) Exact value

100 2.507593 ± 0.000016 2.507599 ± 0.000007 2.50759

200 2.745307 ± 0.001456 2.746076 ± 0.000315 2.74604

300 2.873277 ± 0.007087 2.877765 ± 0.000536 2.87777

400 2.956663 ± 0.016230 2.968180 ± 0.000000 2.96818

500 3.015873 ± 0.027329 3.036800 ± 0.001002 3.03670

600 3.060209 ± 0.039313 3.091950 ± 0.001950 3.09170

700 3.094683 ± 0.051573 3.137915 ± 0.002465 3.13755

800 3.122263 ± 0.063778 3.177185 ± 0.002348 3.17679

900 3.144831 ± 0.075755 3.211338 ± 0.001526 3.21105

1000 3.163642 ± 0.087418 3.241440 ± 0.000000 3.24144

Table 4.9. Approximations and bounds for Il~~~ in the

~tandard nonnal distribution

n Sugiura's bound Bound from (4.65) Exact value

75 5.9689676 ± 0.0031344 5.9706607 ± 0.0000071 5.9706671

80 6.0802196 ± 0.0045483 6.0827371 ± 0.0000000 6.0827371

85 6.1847147 ± 0.0063162 6.1882830 ± 0.0000568 6.1882793

90 6.2831558 ± 0.0084588 6.2880338 ± 0.0000832 6.2880179

95 6.3761356 ± 0.0109903 6.3825878 ± 0.0001029 6.3825623

100 6.4641592 ± 0.0139167 6.4724659 ± 0.0000935 6.4724306

105 6.5476619 ± 0.0172393 6.5581089 ± 0.0001423 6.5580668

110 6.6270211 ± 0.0209553 6.6398926 ± 0.0001337 6.6398545

115 6.7025678 ± 0.0250577 6.7181548 ± 0.0000991 6.7181269

120 6.7745939 ± 0.0295364 6.7931759 ± 0.0000000 6.7931759

97

The coefficients Jj defined by Saw and Chow in their equation (14) for the purpose of approximating S are

related to the Fourier coefficients aj; in fact, it can be shown that af = (2j+l) Jf. It is also of interest to

mention here that Royston (1982) has developed an efficient algorithm for computing the exact and approxi

mate values of expected normal order statistics.

98

Exercises

1. (Ruben (1954); Saw and Chow (1966); Balakrishnan (1984». With aj as given in Table 4.6, show for

the standard nonnal distribution that

n 2k-l 1 _ 1 ~ 2 _ ~ 2 n! (n-l)! Ii S - Ii L ~i:n - L aj (n+J)!(n-J-l)! + ~-l'

i=l j=O where a2j = 0 (j = 0,1,2, ... ) and ~k-l is the error term. If the upper and lower bounds for R2k- 1

* are denoted by R2k_1 and R*2k-l' respectively, then show that

2k+1 } * _ (n-l)(n-2) ... (n-2k-l) { ~ 2 ~-l - R*2k-l - (n+l)(n+2) ... (n+2k+l) 1 - . L aj

J=O which can be evaluated for any given value of n and k.

2. (Balakrishan and Joshi (1985». (i) For positive integers a and b, let the function H(a,b) be as

defined in Theorem 4.3. Then for standardized symmetric distribution with fmite ~7~' show that for

k~O

I (n+l)(n+2) ~n+l:n+2 - c (m+l)(m+2) ~m+l:m+2 - d <p+n(p+2) ~p+l:p+2 k

- l a2j+lb2j+ll j=O

k

{ I (2) ~ 2 }112 { 2 2 ~ 3U 1l3:5 - j~O alj +1 H(n,n) + c H(m,m) + d H(P,p) - 2c H(m,n)

k } 112 - 2d H(p,n) + 2cd H(m,p) - l bij+l '

j=O where

, 11 -1 aj = 0 F (u) u(l-u) 'j(u) du

and bj is as defmed in (4.60), and the optimal values of c and d are as given in Theorem 4.3.

(ii) With Fourier coefficients aj as defmed in equation (4.59), show for distributions symmetric about

zero that for j = 1,3,5, ...

4a' - jU-I) a + (j+l)(j+2) a - j - (2j-3)In(2j-l)(2j+I)In j-2 (2j+I)1/2(2j+3) (2j+5)1/2 j+2

{ C+1)2 .2 } + (2J+i)(2J+3) + (2J-li(2J+l) -I aj

with the convention that a_I = O.

3. (Balakrishnan and Joshi (1985». For positive integers a and b, let

99

F(a,b) = ab(a-I)(b-I) !(a+b-l)(a!b-2)(a+b-3) - i B(a,b) J' where B(a,b) is the comp ete beta function. Then for symmetric distributions with mean 0 ·and

variance I, show that for k ~ 0

k

I ~n-I:n - c ~m-I:m - d ~p-I:p - l ~j+lbZj+11 j=O

< ([ - it ~i+[ t2 (F(n,n) + .,2 F(m,m) + ." F(p,p) - 2c F(m,n)

k }1!2 - 2d F(p,n) + 2cd F(m,p) - l bzJ+I '

j=O where aj is as defined in equation (4.59),

bj = J~ {(n-I)Un- 2(I-U) - c m(m-l)um- 2(1-u) - d P(P-I)uP- 2(1-U)} cplu)du

= {n bj,n_1 - (n-I) bj,n} - c{m bj,m_1 - (m-I) bj,m} - dip bj,p_1 - (P-I) bj,p}

with b. as in (4.62), and the optimal values of c and d are given by J,s 2

Copt = (Um,pUn,p - Um,nUp,p)/(Um,p - Um,mUp,p)' with

k

ua,b = F(a,b) - l {a b2j+l ,a_l-(a-l) b2j+l,a} {b b2j+l,b_l-(b-l) b2j+l ,b}· j=O

4. For the case pf the standard normal distribution, by discarding the beta terms in the inequalities in

Exercises 2 and 3, compute the bounds and approximation for ~n+l:n+2 for n = 48 (50) 398 by

taking m = 248, P = 398 and k = 8. Compare these with the bounds and approximations of the same

order obtained by Sugiura's and Joshi's methods along with the tabled values taken from the tables of

Harter (1961) and comment.

5. Let L 2(0,1) and L 2(R) be the spaces of all square integrable functions in (0,1) and the square R

= {(u,v): 0 ~ u, v ~ 1}, respectively. If (CPk(u)}k=O is a complete orthonormal system in L 2(0,1),

then prove that (CPk(u) cP !v)}k,l=O is a complete orthonormal system in L 2(R).

6. Decompose L 2(R) into the following four subspaces:

L2 (R) = {f(u,v)lfeL2(R) and f(u,v) = f(l-u,v) = f(u,l-V)}, e,e

L2 (R) = {f(u,v)lfeL2(R) and f(u,v) = f(1-u,v) = -f(u,I-V)}, e,o

L2 (R) = {f(u,v)lfeL2(R) and f(u,v) = -f(1-u,v) = f(u,I-V)}, o,e

L 2 (R) = {f(U,v) I feL2(R) and f(u,v) = -f(l-u,v) = -f(u,I-V)}. 0,0

100

Let (<Pk(u)}k=O be any complete orthononnal system in L2(0,1) satisfying <Pk(u) = (_I)k ~(l-u). 00 2 00 2 (

Then, show that (<P2k(u) <P2tv) }k,bO e Le,e(R), (<P2k(u) <PU+ 1 (v) }k,f=O e Le,o(R), <P2k+l(u) 00 2 00 2 . d

<P2tv)}k,bO e Lo,e(R), (<P2k+l(u) <PU+I(v)}k,t=O e Lo,o(R) and that each subsystem IS complete an orthononnal in the corresponding subspace.

7. Suppose that F(x) is the cdf of a random variable X and that E(X2) is finite. Then, show that

F-I(u)F-1(v) e L2(R). If, further, the distribution is symmetric about 0, then show that F-1(u)F-1(v)

2 e Lo,o(R).

8. (Sugiura (1964». (i) Let (<Pk(u)k=O' with <PO == 1, be a sequence of complete orthonormal functions in

L 2(0,1). Then for an arbitrary standardized distribution with mean 0 and variance I, show that for I ~ i<j~nandt~1

t

Illij:n-i l akat[bk,t+ bl,k] I k,l=1

t ~ {I _ l a2 a2}112 {B(2i-1 ,2j-2i-l,2n-2j+l) _ B(2i-l,2n-2i+l)

k,t=1 k t 2[B(ij-i,n-j+I)]2 2[B(i,n-i+I)]2

t _ B(i+j-l,2n-i-j+l) _ B(2j-l,2n-2j+l) + 1 _1 \' [b A- b ]2}112,

B(I,n-1+1)BG,n-j+1) 2[B(j,n-j+I)]2 :if k,~=1 k,t l,k

where

II -I ak = 0 F (u) <Pk(u) du,

1 II i-I ,j-i-l n-j bk,t = B(IJ-l,n-j+1) u (v-ur (l-v) <Pk(u)CPtv) du dv O<u<v<1

and

_ r(a) r(b) r(c) B(a,b,c) - I'(a +b+c) , (a,b,c > 0).

(ii) For an arbitrary standardized distribution with mean 0 and variance I, show that for I ~ i < j ~ n

Ill ... I ~ {B(2i-I,2j-2i-I,2n-2!+I) _B(2i-I,2n-2j+l) _B(2j-I,2n-2j+l)

IJ.n 2[B(i,j-i,n-j+I)] 2[B(i,n-i+I)]2 2[B(j,n-j+I)]2

B(i+j-I,2n-i-j+l) }I12 _ s: - B(1 ,n--l.+1) BG,n-j+1) + I - u (say),

and that equality holds if and only if

F-1(u) F-I(v) = + 1 1 + g(u v) _ u (l-u) + v (I-v) { i-I n-i i-I n-i

- 0 ' 2 B(I,n-1+1)

ui-1(l-u) n-j + J-l(l_V)n-j} - 2B(j,n-J+1) ,

where _ 1 i-l ,j-i-I n-j

g(u,v) - 2B(IJ-i,n-J+1) u (v-ur (I-v) ,0< u < v < 1

101

_ I i-I ,j-i-l n-j - 2B(I,j-l,n-j+ 1) v (u-v r (l-u) ,0 < v < u < 1.

9. By defining the function I(a,b,c,d,e) as

I(a,b,c,d,e) = fJ ua-1vb-I(l_v)c-l(v-u)d-I(I_u_v)e-1 du dv

O<u<v<l u+v<l

for positive integers a,b,c,d,e, show that

b-I c-l

I(a,b,c,d,e) = l l [bk"l] [ci] B(d+k,e+l,a+b+c-k-l-2) 2-(a+b+c-k-l-2)

k=O l=O and

I(a,b,c,d,e) = I(a,c,b,e,d).

10. (Sugiura (1964». (i) Let {<I>k(u)}k=O' with <1>0 == I, be a sequence of complete orthononnal functions

in L 2(0,1). Then for an arbitrary symmetric distribution with mean 0 and variance I, show that for I

~i<j~nandt~O

t

I Ili,j:n - i l a2k+1 ~l+ I [b2k+I,21+ I + b21+I,2k+l) I k,l=O

t t

~ {l- l ~k+1 a~l+lp/2 {KI -~ -~ l [b2k+I,2l+l + b21+I,2k+I)2p/2, k,l=O k,l=O

where

K = B(2i-I,2j-2i-I,2n-2j+I)+B(n+i-~,n+i-j,2j-2i-l),

I 8[B(i,j-i,n-j+I)]

~ = I Z {1(2i-I,n-j+I,n-j+lj-i,j-i) 8[B(ij-i,n-j+I)]

+ 21(n+i-j,i,n-j+I,j-i,j-:-i) + 1(2n-2j+ l,i,i,j-ij-i) },

ak and bk,l as defined in Exercise 8, and I(a,b,c,d,e) as defined in Exercise 9.

(ii) Por an arbitrary symmetric distribution with mean 0 and variance I, show that for I ~ i < j ~ n

Ill ... I ~ I {B(2i-I,2j-2i-I,2n-2j +l) l,j.n 2.fZ B(i,j-i,n-j+l) +B (n+i -j ,n+i-j ,2j -2i-l)

-I (2i-I,n-j+1,n-j+I, j-i, j -i) -21 (n+i-j ,i,n-j+I,j-i,j-i)

1(2 2· I ..... . )}1/2 - n- J+ ,1,1,]-1,]-1

* = B (say),

and that equality holds if and only if

* * p-I(u) p-I(v) = ± ~* {g (u,v) - zg (u,l-V)},

where

* _ I { i-I n-j n-j i-I} j-i-I 0 I g (u,v) - 4B(I,j-l,n-j+I) u (l-v) +u (I-v) (v-ur , < u < v <

102

1 {i-l n-j n-j i-I} ,j-i-l = 4B(IJ-I,n-J+l) (l-u) v +(I-u) v (u-vr ,0 < v < u < 1.

11. (Sugiura (1964». Let (<Pk(u)}k=O' with <Po == I, be a sequence of complete orthononnal functions in

L 2(0,1). Then for an arbitrary symmetric distribution with mean 0, variance I and E(0) finite, show

that for 1 ~ i <j ~ n and t ~ I

t

I (2,2) [(2) + (2)] + II\' * * [b b ] I Ili ,j: n - Ili: n Ilj : n - 2" L a2k~l 2k,2r 2l,2k

k,l=1 t t

{ _-4 2 *2 *2}1!2 { 1 [ ] 2}1/2 ~ [E(x ")-1] - 2 ~k ~l Kl + ~ - IS + 1 - 4" 2 b2k,2l + b2f.2k ' ~~ ~~

where

* r1 -1 2 ak = JO {F (u)} <Pk(u) du,

bk,l is as dermed in Exercise 8, Kl and K2 are as dermed in Exercise 10, and

IS = B(2i-I,2n-2i+l) + B(n,n) + B(2j-l,2n-2j+1) + B(n,n) 4[B(i,n-i+1)]2 4[B(j,n-j+I)]2

+ B(i+t1,2n-i-tl) + B(n+i-j,n-i+j). B(l,n-H ) B(J ,n-J+l) ,

for 1 ~ i < j ~ nand t ~ 0,

t

11l~:# -Ilj:n - 2 a;k+2 ~l+1 b2k+2,21+11 k,l=O

t t

~{E(0)-I- 2 a;~+2~l+d1!2{K4-K5- 2 b~k+2,21+1P/2, k,l=O k,l=O

where ak and bk,l are as defined in Exercise 8,

K = B(2i-l,2j-2i-l,2n-2~+I) _ B(2j-l,2n-2j+l) - B(n,n) 4 4[B(i,j-i,n-j+I)] 2[B(j,n-j+I)]2

and

K = I(2i-I,n-j+I,n-j + I,j-ij-i) - I(2n-2j+I,i,ij-ij-i). 5 4 [B(ij-i,n-j+I)]2 '

for 1 ~ i < j ~ nand t ~ 0,

t

where

IIlr.j~~ -Ili:n - 2 ~k+l a;l+2 b2k+I,21+21 k,l=O t t

~ {E(0) -I - 2 a~k+l a;~+2} 1/2 {K6 + K5 - 2 b~k+I,21+2P/2, k,l=O k,l=O

K = B(2i-l,2j-2i-l,2n-2~+I) _ B(2i-l ,2n-2j +1) - B(n,n) 6 4[B(i ,j-i,n-j +1] 2[B(i,n-i+I)]2'

103

12. (David and Johnson (1954». (i) Following the notations of Section 2, for odd values of nand m =

(n+1)/2 show that

E(X ) ~ G + 1 G" + 1 G i v, m:n m &[ri+2) m 128(n+2)2 m

E(X - ~ )2 ~ 1 (G,)2 + 1 {2G" G'" + (G")2} m:n m:n 4[ri+2) m 32(n+2)2 m m m'

E(Xm.n - ~m.n)3 ~ 3 2 (G:rl G~, . . 16(n+2)

E(Xm:n - ~m:n)4 - 3{E(Xm:n - ~m:n)2}2 ~ 1 {3(G' G")2 + (G,)3 G'" _ 6(G' )4},

16(n+2)3 m m m m m

E(Xm .n- ~m·n)3 3 G~ ~1(Xm:n) = {E(Xm : ~-Jlm: n;2}3!2 ~ 2~n+2 0;;;-'

and

~ (X ) = E(Xm:n - ~m:n)4 - 3{E(Xm :n - ~m:nh2 2 m:n {E(X _ II )2}2

m:n t"'m:n G'" rON 2

~nk{o;+3lG!] -6}.

(ii) If x' rv is the standardized upper l00a.% point of the distribution of the median, that is, m,n,U.

Pr [ Xm:n - ~m:n > x' ] - a {E[X _ II ~2}ln m,n,a - ,

m:n t"'m:n then by using the inverse E geworth expansion show that

G" x ' ~ Z + _1_ m (z2 - 1) m,n,a (a) 4.[ii+! -u; (a)

+~{2~-6) Ha)-3z(a)] -3~t Z(a)}'

where z(a) is the upper l00a% point of the standard normal distribution.

13. (David and Johnson (1954». For the Uniform (0,1) distribution, denoting Uk for ~ i~1 (ui:n - nk) show that the moments of Uk are given by the following equations:

E(uk) = 0,

E(I{) = k+i (2(n+1)(2k+l) - 3k(k+1)}, 12k(n+ 1) (n+2)

E(u3) = (k+1 )2(n-k+1)(n-k) k 2k(n+1)\n+2)(n+3)

and

E(U:) - 3{E(I{)}2 = 3 4 k+l {24(n+1)3 (2k+l)(3k2+3k-l) 120k (n+1) (n+2)(n+3)(n+4)

- 40(n+1)2k(k+1) (13k2 +13k+1) + 300 (n+1) k2(k+1l(2k+1)

- 225 k3 (k+1)3 - 5 k(~:i) [2(n+1) (2k+1) - 3k(k+1)]2}.

104

14. (Mathai (1975». Let '0 == 1. '1"2 •... be an orthonormal system in L2(0.1). Then for an arbitrary

standardized distribution with ~ = 0 and (J = 1 and for 1 S il < i2 < ... < ik S n. by denoting

.. . .. . r(i l ) r(i2-i l) ... r(n-ik +1). B = B(11'12 -11' .... lk -lk_l .n -lk + I) = J'(n+l)

rl 1 aA = JO ~ (u) 'A(U) du

and

show that for t ~ 1

k k k k

+ ... -k l ···l l ···l r l=1 rk- l =1 sl=1 ~_I=1

B(lr .lr -l r ..... lr -lr .n-lr +1) B(IS .IS -IS ••••• n-is +1) 1 2 1 k-l k-2 k-I 1 2 1 k-l t t 21/2

- l ... l {k (r l r ) bAr •...• Ar }] • Al=1 ~=1 1'···· k 1 k

where l denotes the sum over all permutations of the integers 1.2 •...• k. (rl ···,rk)

15. (Mathai (1976». Let the parent distribution be symmetric about zero. Let CPo == l.cpl' cp2 •... be a

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complete orthonormal system in L 2(0,1) such that <I>/u) = (-l~ <l>P-u), j = 1,2,... Then for 1 :5: il < i2

< ... < ik ::;; n, show that for t ~ 0

t t

I E(X. . X. . .. x . ) - 1: ... \' an + 1···a2A +1 11·n 12·n lk·n L 1 k

A1=0 ~=O

where

*2 {I }2 b2A1+ 1, ... ,2Ak+ 1 = Kf ( 1: ) b2A +1, ... ,21. +1

r 1' ... ,rk r 1 rk

and, as before, ( 1: ) denotes the sum over all permutations of the integers 1,2, ... ,k. r1,···,rk

16. (Chu and Hotelling (1955); Siddiqui (1962». (i) Let X be a random variable with cdf F(x), pdf f(x),

and a unique median which may be taken as zero without loss of any generality. Suppose that m(z),

the inverse function of z(m) = 2F(m)-I, is for -1 < z < 1 uniquely defined and equal to a convergent

series of powers of z; let

m(z) = a1z[1 + ~ cp zPJ. p=1

Then, show that for k = 0,1,2, ...

E[X~!1:2n+l] 2k 1

a1 B(n+l,k+z) [2k+l (2k+l)(2k+3) ] = 1 1+ 2n+2k+3 b2(k) + (2n+2k+3)(2n+2k+5) b4(k) + ...

B(n+l,Z)

and

. 2k+l 3 2k+l (2k+l) a 1 B(n+l,k+Z) [ 2k+3

E(X 1 2 1) = 1 c1+ 2n+2k+5 b3(k) n+ : n+ B(n+l,Z)

(2k+3)(2k+5) ] + (2n+2k+5)(2n+2k+7) b5(k) + ... ,

where b2(k),b3(k),b4(k), ... are expressible in terms of k,c1'c2' ... ; for example,

2 b2(k) = k(2k-l)c1 + 2kc2,

1 3 b3(k) = 3 k (2k-l)c 1 + 2kc1c2 + c3'

etc.,

(ii) If Cp/pk -l constant (which may be zero) for some integer k, then the series in (i) are convergent

for all integers n ~ no.

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(iii) If f(x) is symmetrical about O. then show that c1 = c3 = ... = 0 and also that E(X~!r~2n+l) = O.

whenever it exists. for k = 0.1.2 •...

(iv) Let Fl and F2 be two different symmetric populations. Let m1(z) and ~(z) correspond to Fl

and F2• respectively. Further. let Xn+I:2n+1 and Y n+I:2n+ 1 be the medians in samples of size 2n+1

drawn from the populations Fl and F2• respectively. If ml(z) S; ~(z) for 0 < z < I. where strict

inequality holds over an interval of non-zero length. then show that for k = 1.2 •...

2k 2k E [Xn+l:2n+l) S; E (y n+I:2n+l)-

17. (Tiku and Malik (1972». Let Xi:n (I S; i S; n) be the i'th order statistic in a sample of size n from an

arbitrary distribution. Using the three-moment chi-square- and t-approximations given by

2 Xf ~ (Xi:n +a)/g and \r = (Xi:n +d)/h.

where

f = 8/131 (the degree of freedom of X2). g = ~ 1lJ/J.l.2 and a = gf - J.l.i (J.l.3 > 0)

and

v = 4 (132-1.5)/(132-3) (the degree of freedom of t).

h = ~ J.l.2(1 - ~) and d = - J.l.i.

with J.l.i as the mean. J.l.k as the k'th central moment of Xi:n, 131 = J.I.~/J.l.i and 132 = J.l.JJ.I.~. verify by

computations that the t-approximation provides reasonably good approximations for the percentage

points and the probability integrals of Xi:n for small values of 131' say < 0.01; similarly, if 131 ~ 0.01,

verify that the chi-square approximation provides reasonably good approximations as long as 132 does

not deviate from the 132- value of the X2, viz., 3 + 1.5131• Note that if J.l.3< O. we may replace X7 by

2 1- Xf" Remark: You may do the comparisons for the normal, Student's t. logistic, gamma, Weibull and

Pareto distributions. It should be pointed out here that these approximations may not be satisfactory

for the extreme of the lower tail.

18. (Tadikamalla (1977». (i) Let Xi:n (1 S; i :;; n) be the order statistics obtained from a random sample

of size n from the four parameter Burr distribution with c.d.f.

F(x) = 1- {I + [T]c}~. x ~ a.

where d,c > 0 are the shape parameters, b > 0 is the scale parameter and a is the location parameter.

Then show that for k ~ I

i-I k _ n! k t ,j (i-l)! . k. k

E(Xi:n -a) - (1-1)! (n-i)! b d L (-IT J!(1-1-J)! B(nd - Id + d - c + Jd, I + c), j=O

where B(.,.) is the complete beta function.

(li) By determining the parameters d.c.b and a of the Burr distribution that approximates the gamma

distribution with pdf

* -x a-I f (x) = e x tr(a), x ~ O. a > O.

for any given a by matching their fIrst four moments (mean. variance, skewness and kurtosis),

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compute the approximate values of the first four moments of all order statistics from the above

gamma distribution for sample sizes up to 10. By comparing them with the exact values given by

Gupta (1960), comment on the accuracy of this approximation procedure.