Recurrence relations between product moments of order statistics

Journal

of Statistical

Planning

and Inference

8 (1983)

175

175-183

North-Holland

RECURRENCE RELATIONS BETWEEN MOMENTS OF ORDER STATISTICS

PRODUCT

A.H. KHAN, Saud PARVEZ and Mohd. YAQUB Department of Statistics, Aligarh Muslim University, Aligarh-202001, India Received

28 February

Recommended

1982; revised

by N.L.

manuscript

Abstracf: A general result for obtaining statistics duct

is established

moments

exponential,

recurrence

16 August

relations

and this result is used to determine

of some Pareto,

received

1982

Johnson

doubly

power

truncated

function

distributions.

and Cauchy

between

product

the recurrence The examples

moments

relations considered

of order

between

pro-

are Weibull,

distributions.

AM.7 Subject Classification: 62G30, 62699. Key words andphrases: Truncated tion and Cauchy

and non-truncated

Weibull,

exponential,

Pareto,

power func-

distributions.

1. Introduction

Some recurrence relations between product moments of order statistics are given by Malik (1966,1967), David (1970) and Balakrishnan and Joshi (198 1,1982) among others. In the references cited, a particular distribution is considered and the relations are obtained. In this paper, we establish general results for obtaining the product moment of the j-th power of the r-th order statistic and the k-th power of thes-th order statistic. Then these results are utilized to determine recurrence relations for doubly truncated Weibull, exponential, Pareto, power function and Cauchy distributions. To this end, we proceed as follows: Let X1:nlX2:n5... IX, :n be the order statistics obtained from a continuous distribution function (df) F(x) and probability density function (pdf) f(x). The joint is given by pdf of X,:, and X,:, (1 ~r
--oo
n! Cr,s:n= (r- l)!(s-r-

0378-3758/83/.$3.00

0

1983, Elsevier

l)!(n-S)!

Science

.

Publishers

B.V. (North-Holland)

(1.1)

176

A.H. Khan et al. / Recurrence relations between product moments

Let then a(_i,k) = r,S:”

c

r,s:n

. {1

s s -F(Y~TW-~Y)

m mx~yk{F(x))‘-yF(y)-F(X)}S-‘-~ dy dv.

The pdf in case of truncation

s, where

from both the sides is

Q,sxsP,,

s

(1.2)

(1.3)

PI

+f(x)dx=Q

and

mf(X)dx=l-P. s P,

P and Q are assumed to be known (Q < P) and Qi and P, are functions of Q and P. For simplicity, we use f(x) and F(x) for truncation case as well. a(‘*!) r,s.n will be denoted as a I-,.Y.tl*

2. Recurrence relations for product moments In case of truncation ,(A? = c r,s:n r.s.n

from both the sides, p, p,

x’yk{F(x)}‘-

. {F(y;li(x))S-‘-l(l Theorem 1. For llr
’

ss -F(y))“-Sf(x)j-(y)dydx.

(2.1)

and j,k>O,

a(j.f) _ a(i.k) r,s-l:fl =c,fls-,:nk r,s.n

p, PI x’yk-’ {F(x))‘- ’ s s

. (F(y)-F:);s-‘-I{1

-F(y))“-S+lf(x)dydx

(2.2)

where cTs-l’n=(r_

l)!(s_r_

n! l)!(n_s+

C&s-1:n l)! = (s-r1) .

Proof. We have .(L k) _ .(i. k) =c&-1:n r,s-l:ll r,S:”

p, p, s

xju” {F(x)}‘- ’

s

- (F(y)-:(~;]‘-~-~(l

-F(y)}“-’

.{(n-r)F(y)-(n-s+l)F(x)-(s-r-l)}f(xlf(y)dydx. (2.3)

A.H.

Khan et al. / Recurrence

relations

between

product

moments

177

Let h(x,y)=-{F(y)-F(x))S-‘-l{l

-F(y)}“-s+l,

(2.4)

then y

= {F(y)-F(x)}S-r-2{

1 -F(y)}“_S

. {(n-r)F(y)-(n-s+

l)F(x)-(s-r-

l)}f(Y).

(2.5)

Putting the value of (2.5) in (2.3), we get p,

,(i. !I _ ,(i, k) r.s.n r,s-1 :n =&1:n

p,

xj{F(x)}‘_ ‘j-(x)

s PI

yk -fj

Mx, Y) dy

IS x

dw. 1 (2.6)

Now, in view of (2.4), PI yk i sX

h(x, y) dy = k “yk-’ {F(y)-F(x)}‘-‘-‘{ ix

1 -F(y)}n-S+’

dy. (2.7)

Substituting Corollary

(2.7) in (2.6), the required expression is obtained. 1. For l
k)

r,r+

1 :n

andj,k>O,

=a$k)+c,:.k

~‘y~-~(F(x)}‘-~{l

-F(y)}“-‘f(x)dydx (2.8)

where C

r,r+l:rl c. =----=r.n (n-r)

Proof.

n! (r-l)!(n-r)!

Putting s= r+ 1 in Theorem

*

1 and noting that

cz(ik) =E(XJ’,.X,k:n)=E(X~,+k))=al(n+k), r.r:n

(2.9)

we get the desired result. Corollary 2. For n > 1 and j, k> 0, &i,

k)

?I-l,tl:n

=

a;:,:), + n(n

-

1)k

p, p, ss

x’yk-‘{F(x)}“-2{1-F(y)}f(x)dydx.

Q,

Proof.

Put r= n - 1 in Theorem

Theorem 2. For 1 sr
x

(2.10)

1, to get the result.

and j>O,

&.I)) = &.O) ,(A = . . . =(r (4 0) r,s.n r,S-l:n r,r+1:n= r:n*

(2.11)

178

A.H. Khan et al. / Recurrence relations between product moment3

Proof. From relation (2.3) and (2.6), with k=O,

s PI

_ .(j. 0) r,s-1:n

x’ {F(x)} r- ‘f(x)

QI

But

s

pNwx,Y) ____ dy=h(x,P,)-h(x,x) ay

X

=0,

in view of (2.4).

Therefore, (LO) (LO) U,0) a,,s:n=a,,s-l:n=“‘=ar,,+I:n’

(2.12)

But a(j,O)

r,r+l:n

xj(~(x)}‘-‘{1-F(y)}“-‘-‘f(xlf(y>d~dx.

=cr,rt1:n

(2.13) Integrating

(2.13) by parts with respect to y, we get

The usual technique in establishing the recurrence relations will be to express { 1 -F(y)} as a function of y and f(y). It is interesting to note that the relations for specific distributions are obtained by just substitution.

3. Examples 3.1. Doubly truncated WeibuN and exponential The pdf of the Weibull distribution

distributions

is given by

-2

f(x) =

Here Qf= -log(l Q2=-

px;;‘; -Q), I-Q

P-Q

)

-log(l

Pf= -log(l and

- Q)SX~I

-P).

-log(l

-P),

p>O.

(3.1)

Let

P2=eQ,

then {l-F(y)}=-_q+~yl-ilf(y).

(3.2)

A.H. Khan et al. / Recurrence relations between product moments

Putting this value of { 1 --F(y)}

179

in (2.2), we get p, p,

o(i.? _ o(.i,k) r,s.ll r,s-l:?l =c;s-i:nk

=-pzP

X’y~-l{F(x)}‘-‘{F(y)-F(x)}S-r-l

s Q, sx

1 n-s+1

X’yk-l{F(x)}‘-’

cr,s:nk

k

+ &l--s+

1)

x’yk-“{F(x)}‘-

G,S:ll

l

That is, &, W = ,(i r,s:n

4

r,s-l:n

+

From Corollary

k p(n-s+

&f-p)

15r
1) rVs.n ’

(3.3)

1, for s = r+ 1, (3.3) reduces to

,(I k) r,r+l:n

= &+k) -- nP2 r.n

{&k)

(n-r)

k + p(n_r) From Corollary

_ &+Q

r,r+l:n-1

,(_Lk-p)

r*r+l:n’

r.n-l

1

llrln-l,nr3.

(3.4)

2, for r= n - 1 and s= n, (3.3) reduces to

&.k) fl-l,n:n

= ayT,k)n - nP2 { p:cY;j 1: n _ 1- ayy!, + k &k-p) n-1,n:ft P

_ I>

n12.

(3.5)

Expression (3.5) could also have been obtained from (3.4) by putting r=n - 1. In substitution, we get a term cry:? n :n_ 1 which is esentially an undefined term. This can be interpreted as E(X, _ , : ,, _‘, Pf ) = Pf&)n _ 1: n _ , , where P, is the upper limit of the Weibull variate. However, we reached at this conclusion after doing actual calculations as given below: ,(i

4

?l-1,tl:ll

= cry:,:), - nP,(n - 1)k

s PI

x’{F(x)}“_2

Q,

(j;’y*-I dy)f(x~~

xj~~-“(~(x))~-~f(x)~(r)

dy dx

dx

A.H. Khan et al. / Recurrence relations between product moments

180

+ k &,k-PI n-1,n:n

P

If we put p = 1 in the above expressions, we get corresponding results for the exponential distribution. For the non-truncated case one has to put P = 1, Q = 0. In case of the doubly truncated Weibull distribution, recurrence relations for a$?; are available in Khan et al. (1983). Expressions for exact and explicit product moment with j= k= 1 can be obtained in Lieblein (1955). In case of j= k= 1, for the Weibull distribution, 1

nP2

or,s:n=or,s-l:n

--

Iar,s:n-l

n-s+1

-%,s-l:n-d+

p(n_s+l)

1 sr
distribution,

=%,s-1:n-

($1’ 1 -P)

r,s:n

’

s-rr2.

(3.6)

s-rr2,

(3.7)

this reduces to

(n_s+l) nP2

{(rr,s:n-l-(Yr,s-l:n-l}+(Tr:n/(n-s+*),

1 sr
in view of Theorem 2. From (3.3), it is clear that if kcp, then the power of Y will be negative. So-far, we have not been able to obtain inverse product moments for the Weibull distribution. 3.2. Doubly truncated power function The pdf of the doubly truncated Ua-“Xuf(x)=

distribution

power function

distribution

is given as

1

aQ”“sxsaP”“,

P_Q

a,o>O.

Here Qt = aQ”‘, P, = aPI”. Let P2= -

P

P-Q

and

Q2=-

Q P-Q’

then (1 -F(Y))

=P2-

3Y).

(3.8)

A.H. Khan et al. / Recurrence relations between product moments

Putting the value of (1 -F(Y)} ,(i. f)

from (3.8) to Theorem

181

1, we get

-a~;k_),:n=c,ffS:nk ,

r,s.n

On simplification,

we get

,(j.f) = r,s.n u(n-s+

II

[(n --s + l)ai,$k_),:n + nP2(aci,!) r,s.n _ I - aCi,k_) r,s 1:n-1)19

l)+k

1 sr
,(6 k)

r,r+t:n=

o(n_r)+k

s-rr2.

(3.9)

1,

[(n -

r)a(i+ k, r.n kJ+ nP,(acir,r+l:n-1 lsrsn-2,

n23,

(3.10)

= a(i+ 0 after noting that aCx? r.n ’ r,r.n Similarly for n =s= r+ 1, we get

interpreting a(ck) as discussed in Example 3.1. n l,n:n_l=P~o~?,I:._, For j= k= 1, the relations are available in Balakrishnan and Joshi (1981). The non-truncated cases (P = 1, Q = 0) are discussed by Malik (1967). Reference may also be made to Khan et al. (1983) for the recurrence relations of a:!),,, I= 1,2, . . . . 3.3. Doubly truncated Pareto distribution The pdf of the doubly truncated “aUx-“f(x)=

Pareto distribution

is

1

P_Q

9

a(1 - Q)-““sxsa(1

-P)-I’“,

a, o>O.

Here, Q, = a( 1 - Q)-I’“, P, = a(1 - P)-I’“. Let Q&k!_

then

P-Q

and

P2=fi,

(3.12)

182

A.H. Khan et al. / Recurrence relations between product moments

In view of (3.12) and (2.2),

k = o(n-s+

,Ci,f) + nP2 l~~,j~~~_,-U~,j~k),:n-,}, 1) rSs.n (n-s+ 1)

01

{o(n-s+1)-k}a~~~b=u[(n-s+l)~~;k),:n+nP2(~~~q’,_,-~~~K)1,,_,)], llr
s-r22

and u(n-s+

l)#k.

(3.13)

However, if k= o(n --s+ l), we get from (3.13), (n-s+

1)&k_’ r,s 1.n =nP,(a’“k_) r,s

l:n-1

_ &k)

r,s.n-l

(3.14)

).

Marginal results for k# u(n -s + 1) can easily be seen to be equal to [u(n-r)-k]cr$.:),

:n= ~[(n-r)aji~~)+np,(al’;r),

:“_, -crj!;_k’,)], 11r
(u - k)&ck’ +nP2(P:a~~,:._,-a~~~),_,)], n I,n:n = urcfl;‘_+;),

nr3,

(3.15)

n22. (3.16)

Recurrence relations for j= k = 1 have been studied by Balakrishnan and Joshi (1982). Malik (1966) has obtained these results for P = 1, Q = 0. To evaluate ay;j;if’,, one may require the recurrence relations for (Y,:, (‘) for which we refer to Khan et al. (1983). 3.4. Doubly truncated Cauchy distribution The pdf of the doubly truncated f(x) =

l

--!-

(P- Q)n 1 +x2 ’

Cauchy distribution

is given by

Q,zsx’P,,

where Q, and P, are obtained as given in Section 1. Therefore,

1= (P- QM

(3.17)

+u’)f(u).

In view of (2.2) and (3.17), &k) r,S.n -(YI,~~~),:~=c,T,_,:.~~~(P-Q) . U~Y)-FWY-‘-‘U

= kW - Q) n+l

ra;,jit;j),

p, p,

xjyk-l(l

+y2){F(x)}‘-’

ssQ, x -F(y))“-S+lf(xlf(y)dydw + a(jtk+ 1) 1 r,s:n+l

*

A.H.

Rearranging

Khan et al. / Recurrence

relations

between

product

moments

183

the terms, and replacing n by (n - l), we get &,r+

1) =

r,s.n

(3.18) Similarly, it can be shown that &,k+l) r,r+l:n

n

=

[,(i, k)

lsr(_n-2,

-a~,‘k)ll-(r(;‘;:;;;,

r,r+l:n-1

n/qp_Q)

(3.19) and n &,k+ 1) n-2,“-1 :n = rck(P_

Q)

nr3.

~~~~~,~_]:~_,-a~~~)~_11-rW~~~~~~,:~,

(3.20) If we put j= k = 1, (3.18) reduces (in view of Theorem 2) to &2) r.s:n

=

n

(3.21)

[(Y,,s:n-l-(Y,,,-l:n-ll-ar:n.

71(p _ Q)

For the non-truncated case, put P= 1, Q=O. For the recurrence relation of cr:‘, in this case, one may refer to Barnett (1966). Reference may also be made to Khan et al. (1983).

Acknowledgement

The authors are grateful to the referee for his helpful suggestions.

References Balakrishnan,

N. and P.C. Joshi (1981). Moments

tion distribution. Balakrishnan, distribution. Barnett,

Aligarh

Assoc.

David, H.A. Khan, A.H.,

of order statistics

Order

statistics

from doubly

truncated

power

func-

1, 98-105.

N. and P.C. Joshi (1982). Moments of order J. Indian Statist. Assoc., 20, 109-117.

V.D. (1966).

Statist.

J. Statist.

estimators

statistics

of the location

from

doubly

of the Cauchy

truncated

distribution.

Pareto J. Amer.

61, 1205-1218.

(1970). Order Statistics. M. Yaqub and S. Parvez

1st edn. Wiley, New York. (1983). Recurrence relations

Naval Res. Logist. Quart. 30, to appear. Lieblein, J. (1955). On moments of order statistics 26, 330-333. Malik, H.J. (1966). Exact moments

of order

statistics

from

between

the Weibull

moments

distribution.

from the Pareto

distribution.

of order Ann.

Mafh.

Skand.

statistics. Statist.

Aktuar.

49,

144-157. Malik,

H.J.

Aktuar.

(1967). 50, 64-69.

Exact

moments

of order

statistics

from

a power

function

distribution.

Skand.