On the moments of order statistics from the doubly truncated logistic distribution

)urnal of Statistical Planning and Inference 13 (1986) 117-129 orth-Holland

117

ON THE M O M E N T S OF ORDER STATISTICS FROM THE DOUBLY T R U N C A T E D LOGISTIC DISTRIBUTION N. B A L A K R I S H N A N and S u b r a h m a n i a m K O C H E R L A K O T A Department o f Statistics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2

Received 15 November 1983; revised manuscript received 26 April 1985 Recommended by S. Panchapakesan

Abstract: The paper presents moments and product moments of order statistics from the doubly

truncated logistic distribution. It presents recurrence relations for the single moments of all orders and for all sample sizes. In addition, the recurrence relations in the product moment are also presented for the first order moments. These recurrence relations could be used systematicallyin order to evaluate the means, variances and covariances of all order statistics for all sample sizes. AMS Subject Classification: Primary 62G30. Key words and phrases: Doubly truncated logistic distribution; Order statistics; Moments and

product moments; Recurrence relations.

ttroduction The moments of order statistics in the logistic distribution were examined by Birnaum and Dudman (1963). Tables of the means and standard deviations are presentfor selected sample sizes by them. Gupta a n d Shah (1965), and, independently, arter and Clark (1965), studied the distribution of the order statistics and of the rage for such populations. Explicit formulas for the first four moments are also resented by them. Variances and covariances for these statistics were tabulated by u p t a et al. (1967); these were used by the authors to obtain the B L U E ' s of the parateters. Subsequently, Shah (1966, 1970) established the recurrence relations for the ngle and product moments of the order statistics. The truncated logistic distribution plays a role in a variety of applications, as has ~en mentioned by Kjelsberg (1962). The order statistics and their moments in such istributions have been examined by Tarter (1966). He has given the explicit expresons for the means, variances and covariances in terms of a finite series involving )garithms and dilogarithms of the constants of truncation. Estimation problems ~ve been tackled, a m o n g others, by Kjelsberg (1962). Following the lines of Shah 966, 1970), Balakrishnan and Joshi (1983) have derived recurrence relations satised by the moments of order statistics from a symmetrically truncated logistic 78-3758/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)

118

N. Balakrishnan, S. Kocherlakota / Doubly truncated logistic distribution

distribution. Similar work for the doubly truncated exponential distribution has been carded out by Balakrishnan and Joshi (1984). In the present study we are concerned with the single and product moments of order statistics from the doubly truncated logistic distribution. Recurrence relations are established along the lines of Shah (1966, 1970) and Balakrislman and Joshi (1983). These relations would enable us to calculate the moments of all orders for any sample size.

1. Preliminaries

Let X be a random variable with the probability density function

f(x)=

Q) (1+e-X) 2' 1,_0,

QI
(1.1)

otherwise.

In this case, X is said to have a doubly truncated logistic distribution, with the proportion of truncation on the left and right being Q and 1 - P , respectively. Under this notation, Q l = l n 1 -Q Q'

P P1 = In 1 ------P"

(1.2a)

Let DE - P(1 - P )

p_Q

"

Q2 - Q(1 - Q)

p_Q

(1.2b)

Based on the random sample Xl, ... ,Xn, we consider XI : , - < X 2 : n<- ---<-Xn:, as ,,,(k) the moment E(X~.) and by O[r,s:n the order statistics. Let us represent by '~r:n (r < s) the product moment E(Xr: nXs:,). It can be shown that the pdf in (1.1) satisfies the relations f(x) = (1 - 2Q)F(x)-(P-Q){F(x)} 2 + Q2,

(1.3a)

f(x) =/)2 + ( 2 P - 1){ 1 - F(x)} - (P - Q){ 1 - F(x)}2,

(1.3b)

f(x) =/)2 + ( 2 P - 1)F(x) { 1 - F(x)} + (P + O - 1){ 1 - F(x) } 2.

(1.3c)

and

2. First two moments

Before we present the more general moment relationships, it is necessary to evaluate the particular moments -t,O) ~,(2) " These play a vital role in the determina-r:n, "~r:n tion of the general moments. It should be noted that

iV. Balakrishnan, S. Kocherlakota / Doubly truncated logistic distribution

119

the k-th order raw moment of (1.1). The pdf of Xr:n is readily seen to be fr:n(X)-

1

B ( r , n - r + 1)

{F(X)}r- l { l - F(X)}n-rf(x),

(2.1)

for QI
(2.2)

The following expressions for the moments are readily obtained by integrating (2.2) by parts and using the relations (1.3a) to (1.3c).

A. First moments ff[1:)2 = Q I

a(21:)2=P,

1

11,

(2.3a)

1--~ tQ2{P1-QI} +(1-2Q){P1-a[I:)I}- I ], P-Q

(2.3b)

+ p_----~ [P2 {P1-Q1}

ff~l:) = Q1 + - -

P-O

+

[p2{al:l_Q1} + (1)-

(2P- l){a~l:)~- Q I }

(2P- l){a~l:)2- Q,}-½l,

--(~1:1} +

a3(1:)3 _

-

(2P- 1) {o~20)2_¢z~1:)2} _½] 2 j

(2.3c) (2.3d)

3 [I - 2P2{PI- a~l:)l} - {{3(P + Q - 1)- 2(2P- 1)}a2(1:)3 2(2P- 1) (2.3e)

+ (P + Q - 1)a~l:)21. And for n_> 3, t,~0) k l : n + l = Q l + p _ Q1 [ =

(1)

P2{1~I : n _ l -- Q I } +

(2P-1){a}l:)n - Q1} _ 1] , (2.30

n+ 1 [ 1:'2 {~2q)n_l- - t,~(1) Zl:n_ p_Q 4 (2P-n 1)/atl.~_ail.~}

Finally, for n_3, 2<_r<_n-1,

1]

n(,i-1)

"

1

(2.3g)

120

N. Balakrishnan, S. Kocherlakota / Doubly truncated logistic distribution (1), 0Cr+ 1 : n + l =

n+ 1

[

r(2P-1)

1 nP2 _ ar(D1 :n n-r+l-n-r+ i {a~(~)"-I 1}

{(n + 1)(P+ Q - 1) - r(2P- 1)}ar:n+ (1) 1 - ~ n+l

+ ( P + Q - 1)ur(l__)l:n]

(2.3h)

and

(1),

n+l

~n+J :n+l

n(2P- 1)

[1-nP2{P1 -a~_,.._l} (i),

{(n+ 1)(P+ Q - 1 ) - n ( Z P - 1)}a~l~n+ 1

n+l

+ (P+ Q ~- 1)%_ (i) 1:.J •

(2.3i)

ctff..)2=Q2+

1 [P2(p2-Q2)+(2P-1){ot~z:)l-Q2I-2a~l:)l], P-Q

(2.4a)

ot(22)2=p2

l tQ2{p2-Q2}+(1-2Q){p2-0t}z..)l}-2et}l:)], P-Q

(2.4b)

B. Second moments

1:3

P-Q

[p2{a~2)_Q~}+(2P_l){ a(2) I:2-QI 2 }-or}l:)2], 2 P - 1 {(~,;2.)2-0/}2)2}-

•

a3(2:)_

(2.4.c)

,..(l)] -2:2],

3 [2a20:) _ 2P2{p2_a}2)} 2(2P- 1) - ~ { 3 ( p + Q - 1 ) - 2 ( 2 P - l)}ot(22)3+(P+Q-1)at2)2].

(2.4d)

(2.4.e)

For n_>3,

P-Q

P2{a}2)-'-Q2} +(2P-1){cz}2)-Q2}

n

(2.40

a2 : n (+ l _-2- ~ Z l): n + l 4

4-

n + 1 [ P2 t ~(2)

.:(~

2P- 1 n

Finally, for n > 3 and 2 < r < n - 1 ,

,~(2)

~--yi_1 ~"2:"-'-''' 2

:'-'}

a2(l:)].

(2.4g)

N. Balakrishnan, S. Kocherlakota / Doubly truncated logistic distribution (2) n + 1 [ 2 ar+':n+l-r(~fff'l) Ln - r +

(0 nP2 1 tlr'n" n - r + l

121

(2) {tlr:n-l-~(2-)l:n-l}

1 {(n+ I ) ( P + Q _ 1) _ r ( 2 P - 1 ) } a r (2) :n+ 1 n+l + (P + Q - 1)ar(2)l : ,,/ .1

(2.4h)

n+ l 2ct(npn --n p z { p 2 _ n ( 2 P - l) "

a(2) n+l:n+l

_

rt (2) _l:n_

1

}

1 {(n+l)(P+Q-l)-n(2P-l)}a(2).+ l

n+l

+ ( P + Q - l)an(2_)1:.].

(2.4i)

General recurrence relations

t is possible to derive the recurrence relationships among the single moments of her orders. The following results essentially represent an aspect of such relationps. lation 1. For n>_2, k = 0 , 1,2, ...,

t~(k+l) =Qk+l

I:.+I

i,,,(k+l)_Qlk+l} + p --I Q [p L 21*Zl : n - 1

+ ( 2 p - 1){a~k.• + 1)_ QI*+ 1} - - ~kt +Z ll: n -(k)] .

(3.1)

n

lation 2. For 2
_

•r+1:.+1

n+l

r(2P-1)

[

k+l

o(k)

n-r+l

-r:"

riP2

1 {(n+ 1 ) ( P + Q - 1 ) - r ( 2 P n+l k+ ] +(P+Q _ Do( .,~r-l~)n

•

/..(k+l)

~,(k+l)

n--r+l I~r:n-l-~r-l:"-l} l~/'~'(k+l) l *IIt~r:n+

(3.2)

)of. For relation 1 we start with

.~(k)=Y/ tZl:n

IPI

xk{l_F(x)}n-lf(x)dx

Q~

(3.3)

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N. Balakrishnan, S. Kocherlakota / Doubly truncated logistic distribution

which holds for n >_ 1, k = 0, 1, 2, .... Using (1.3b) in (3.3) and integrating these terms by parts we get

a(k)

1:,-

n rp ~,,,(*+0 _ Q ~ + I } . _ t 2~"1:n-1 +(2P_l){a~*.+l) Q~+I} k+l _ ( p _ Q ) I a (k + l) _ Q ( + 1 } ], t

(3.4)

l:n+l

provided n___2. By rearranging the terms in (3.4), relation 1 is established. To prove relation 2, we start with the definition ~(k)

tZr:n =

l::x

:n

(x) dx.

For 2 _ r_< n - 1, k = 0, 1, 2,..., this integral can be written as

"-B(r,n-r+l)

P2

xk{F(x)}r-l{1-F(x)} n-rdx Qi

+(2P-l)

xk{f(x)}r{l-F(x)}

"-r÷l dx

dQ~

+(P+Q-

1)

I"

xk{F(x)}r-l{1-F(x)}"-r÷2d

dQ,

x]

(3.5)

u p o n using (1.3c). Integrating the right-hand side by parts, we get, after some simplification, (k+ 1) ~.(k+ ~) _(k) Ur:n-k +1 1 nP2{Otr:n_l_t~r_l:n_l} +

r ( n - r + 1) (2P- l~l~(k+- I) ,~(k+ I) - j t ~ r + l :n+ l - - - - r : n + l } _

n+l

-1

lxj',,,(k+ 1) _ t~r-l:n-l} ,,.(k+l) [ • + ( n - r + l)(n-r+2) (-P+Q-,jt-r:n+l

]

n+l

(3.6)

However, the well-known relation [see David (1981), p. 46]

(n - - i ._.,,,_
=(?l+

,,~~~,+ ~) ..¢k+l) } l ] i t Z r : n + 1 --tZr_ 1

u p o n use in equation (3.6) and simplification yields relation 2. N o t e . For n = 1 and r = n, relations 1 and 2 can be shown to become

o f +'

1

- ( k + 1)at~)ll and

lu tP, lp,' + ' - Q~ +l} + ( 2 P - ' l ll'/''(k:

+11) -- QI k +1 }

N. Balakrishnan, S. Kocherlakota / Doubly truncated logistic distribution

123

~(k+ 1) = n (12) P(+n- 1) L[(k+ Hantk).n '-'n+l:n+l • -nP2{P~ +l _(k+ ~ n - l : n1) -l} _

- - ~1

{(n+l)(p+Q_l)_n(2P_l~.(k+l "tlf°~n:n+ )1

n+l

+ ( P + Q - u•u~ . _,(k+ l : .1)j ,] -espectively. ~elation 3. For k = 0 , 1,2, ...,

ark+ O=p~k+l)

1 --[Q2(p~+'-Ql*+l)+(1-

2:2

P-Q

2 Q ) { P ~ +1

- ( k + 1)a~)].

-af*.t')} (3.7)

Relation 4. For n_> 3 and k - 0, 1, 2,..., t~(k+ 1)

.,.(k+ 1).

[- D 1____~) (n + | ~2

t,~(k+ 1)

,,,(k+ 1) /

z:n+l='~l:n+~ +(P-Q) I n - 1 ~'~2:n-l--~'l:n-l~ 2 P - 1 ~,o(k+l ) f f ( k + l ) / ~-z:, -- - , : n J n

k+l

,~(k)]

n(n 1) t~2 : nJ

(3.8)

•

?roof. For relation 3 we start with

,~(k) = IZl:l

S::x~f(x) dx.

(3.9)

Jsing (1.3a) in (3.9) and integrating these terms by parts, we get a(k) 1:1 =

1 k+l

[Q2(PI k + l _ Olk+l ) + (1 - 2 Q ) { P ~ +1 --Ul:l"(k+ -- t~2:2

/l-

1)} (3.10)

]y rearranging the terms in (3.10), relation 3 is established. To prove relation 4 we start with

(k)=n(n-

~2:n

1) Ie~xkF(x){1-F(x)}n-2f(x)dx OQt

[I"

=n(n-1) P2

xkF(x){1-F(x)} n-2 dx

Q,

xkF(x){1 -F(x)} n-I dx

+ ( 2 P - 1) Q,

(3.11)

124

N. Balakrishnan, S. Kocherlakota / Doubly truncated logistic distribution

upon using (1.3b). Integrating the right-hand side by parts and simplifying the resulting expression, we get relation 4. N o t e . For n = 2, relation 2 can be shown to become t2(k+ 2:3 1) = o~k3 "

l)+p_~

2P- 1 "~'(k+ 1)/, + - [P2{ P I k + l -~1:1 2

,,,(k+ 1)

tz2:2

,.,,(k+ 1)}

--tZl:2

k+ 1 .~(k)] 2 Iz2: 2J-

4. P r o d u c t m o m e n t relationships

The joint pdf of Xr:.,Xs:n, 1 <_r
r-

I{F(y)-F(x)}S-r-I{1-F(Y)}

n - s ~f(X )f(Y),

(4.1)

Ql
where /"(n + 1)

Cr, s, n = F ( r ) F ( s - r)F(n - s + 1) 4.1. Basic results

For special values of r, s and n, it is possible to derive the following results directly: 3

a"2:3=a~2)3+2(P-Q) [2P2{

p

(1)

lt~l:l--t~[2.-)l}

+ ( 2 P - 1){ al,2: 2 - a ~2..)2} --Wl~'(1:21 l) and 3

2 ( P - Q)

r,~(1) , ~ f,~(2) of(l) } Lt~2:2 -- , ~ 2 " l t ~ l : 1 -- Q1 1:1_

+(2Q-l){a2t2)2- al,2:2}]. For arbitrary continuous distributions, we have the relation tq,2:2=a2:l [Govindarajulu (1963)]. Recurrence relations connecting continuous product moments are sufficient to fill the product moment matrix. This follows from the result [David (1981), p. 48] (r--1)ar, s:n+(S--r)ar_l,s:n+(n--s+ l)ar_l,s_l:n=ntlr_l,s_l:n_l . (4.2)

To this end, we have the following recurrence formulas:

N. Balakrishnan, S. Kocherlakota / Doubly truncated logistic distribution R e l a t i o n 5. For

125

l <_r<_n-2,

(2) (/7 + 1) [ nP2 (2) a~,r+l:,,+l=ar:n+l+(n_r+ l)(p_Q) Ln_r{ar, r+l:,,-1-ar:,,-l}

• a(2) 1 fr(l)n]. +(2P-1){fr, r+,:nr:nJ/ n--r R e l a t i o n 6. For

(4.3)

2<_r<_n-1, (2)

ar+l'r+2:n+l=ar+2:n+l

nt

n+l

(r+ I ) ( P - Q )

[1

(]) fr+l:n

l

nQ2 {~r~:)_l_fr_l,r:n_l } r "1

(1 Proof.

-

2 Q ) { f r(2) +l:.-

air, r+l : n } |J

(4.4)

Relation 5 can be established by considering the equation, for 1 _ r ~ n - 1, (1) _ fr:n-E(Xr:nX~r+ l:n) Pl

I

=Cr,r+l,n x{F(x)}r-lf(x)D(x)dx,

(4.5)

dQt

where

D(x) =

{ 1 - F 0 0 } n- r- If(y) dy.

(4.6)

x

Using (1.3b) in (4.6) and integrating by parts we get

D(x) =P2[Y {1- F(x)}n- r- t l~' + (n - r -1) I~' y {1- FO')}n- r- 2f (y) dy ] +(2P-1)[y{l-f(y)}n-r l~'+(n-r) I~'Y{1-F(Y)}n-r-lf(y)dy ] -(P-Q)[y{1-F~)}n-r+l l~'+(n-r+ l) I~'y{1-F~)}n-rf~)dy ]. (4.7) For 1 <_r<_n-2, we can replace D(x) by

D(x)=P2[-x{1-F(x)}n-r-l +(n-r-1) I;'y{1-F(y)}n-r-2fO')dy ] + (2P-1)[- x{1- F(x)}n-r + (n- r) I;'y{1- FO')}n-r- lf(y) dy] -(P-Q)[-x{1-F(x)}n-r+l +(n-r+ l) I['y{1-F~)}n-rf~)dY ], (4.8)

N. Balakrishnan,S. Kocherlakota/ Doubly truncatedlogisticdistribution

126

in equation (4.5). Integration of the resulting expression and subsequent rearrangement of the terms yields (4.3).

Note.

If in (4.7) we set r = n - 1, the value of

D(x) becomes

[

D(x)'-P2(PI-X)+(2P- 1) - x { I - F ( x ) } +

-(P-Q)

[

:

yf(y)d

y]

x

]

(2) (1) 1){an_l,n:n-an_l:n} -otn_l:n],

n>_2.

-x{l-F(x)}2+2

y{l-FO,)}fO,)dy

.

This yields the recurrence relation (2)

0~n- l,n :n+ 1 "- ~ n - 1 :n+l "at"

(n+ 1)

2 ( P - Q)

+(2P-

(4.9)

Relation 6 can be proved by considering, for 1 ___r _ n - 1, ct(l)

o r+l:n=E[Xr:nXr+l:n]

= Cr, r +

1,~Ie'Y[ 1 -

F ( Y ) ] ~ - r - ~f ( Y ) E ( Y ) d y

(4.10)

OQ,

where E(y) =

IY

{F(x)} r- if(x)

dx

Q, =Q2

f'

J"

{f(x)} r dx

{ F ( x ) } r-~ d x + ( 1 - 2 Q )

Q,

-(P-Q)

I'

QI

{F(x)} r+l dx

(4.11)

QI

upon using (1.3a). Integration by parts leads to Q-(r-

1)

y x{F(x)}r-2f(x)d Q,

+(1-2Q)[x{F(x)}rIQ -rIQIX{F(X'}r-lf(x) dx]

r" I"0.1-(r+, I"01 Thus, for

2<_r<<_n-1, we have

(4.12)

N. Balakrishnan,S. Kocherlakota/ Doubly truncatedlogisticdistribution

127

E(Y) = Q2 [ Y {F(y) } r- l - (r-1) lY x {f(x) } r- 2f (x) dx ] Q, + (1- 2Q)[y{FO')} r- r lY x{F(x)} r- lf(x) dx] Q~

-(P-Q)[y{F(y)} r+l -

x{F(x)} rf(x) d x] .

(r+ 1) IY

Q~

(4.13)

Substituting for E(y) from (4.13) in (4.10) and rearranging, we get the relation 6. Notes.

(i) I f r = n - 1,

we have

_ tv(2) n+l £ln'n+l:n+l--'*n+l:n+14 n(P-Q)

~Z i

[~_1

(1) an:n

l:n-l--Oln-2~n-l:n-l}

-(1-2Q){atn2)-ct,_l,,:,}],

n>_3.

(4.14)

(ii) For r = 1 in (4.12), we have

E(y)=Q20,-QI)+(1-2Q)[yF(y) - f Q y xf(x) dx 1 - (P - Q ) [ Y {F(y) } 2 - 2 I ;, xF(x)f (x) dx ].

(4.15)

Using this in (4.10), when simplified, we have °~2'3 : n + 1= 0%(2:)+1 + 2 (n+ ( P - 1) Q)

[a(21:)n_nQ2{otl2)n_l_Qzot~!)n_l} (4.16)

- (1 - 2Q){ a2c2)n- al, 2 :. }].

4.2. General results Although, as pointed out in an earlier section, it is possible to derive the covariance matrix from the recurrence relations, we present the general formulas relating to covariances: Relation

7. F o r

1 _ r _ n - 2, ( n + 1)

ar,. :.+ 1= at, n-l :.+1 + 2 ( / ' - Q) + (2PRelation

1 ) { a r . , : , --

8. F o r s - r___ 2 a n d s_< n - 1,

p triP2{

(l)

10lr:n-l--17lr,n-l:n-1}

(i) 1 Glr.n-l:n } -- Olr:n."

(4.17)

N. Balakrishnan, S. Kocherlakota / Doubly truncated logistic distribution

128

(n + 1)

[

nP 2

r,s:n+l--Or,s-I : n + l +(n-s+2)(P-Q) (n-s+ 1) {Ogr's:n-I--ar's-l:n-l} +(2P-

1){Otr,s:n-Otr, s _ l : n } - ( n _ s + 1)

: J"

(4.18)

R e l a t i o n 9. For s > 3, ~2,s+ 1 :n+ 1 = ~3,s+ 1 : n + 1 ~

(n+ 1) (1) ~ ,,,0) [¢t~: n - n Q z { Otl,s_ l : n_ l - ~ l , ~ s _ l :n_ l } 2 ( P - Q)

- ( 1 - 2Q){az,~:,, - al,~:n }].

(4.19)

R e l a t i o n 10. F o r r ~ 2 a n d s - r _ 2,

(n+l) a~+l,~+l:.+~=ar+z.+l:.+l+(r+~ZQ

[~,a,(O )

-~:.

n Q z {ar, s - l : . - X - - a r - l , s - l : n - l }

r

-(1-2Q){ar+l,s:n-ar, s:n}]. 4.3.

Some

(4.20)

comments

Recursive computation of the moments of order statistics is not new and has been carried out for several distributions like gamma, exponential, logistic, double exponential, etc. In particular, for the double exponential distribution, Govindarajulu (1966) has made use of some recurrence relations for the evaluation of means, variances and covariances. The error propagation of this computational procedure has also been investigated by him. It should be noted, however, that these recurrence relations involve combinatorial terms as coefficients which increase with sample size. But in our case and also in the case of exponential, gamma and logistic distributions, the recurrence relations involve only simple algebraic operations and just make use of the values of the mean and the variance of the underlying distribution which are usually known exactly. Hence starting with these values, one could perform the necessary calculations at a high precision on a computer and evaluate the moments of order statistics to the desired level of accuracy, at least for small and moderately large values of the sample size. For similar comments and also a computational evidence, one could refer to Barnett (1966), Govindarajulu (1966), Joshi (1978). Moreover, we have also performed some computation of the moments of order statistics from the doubly truncated logistic distribution making use of-these recurrence relations and, if necessary, these tables could be added for a better understanding of the application of these recurrence relations.

N. Balakrishnan, S. Kocherlakota / Doubly truncated logistic distribution

129

Acknowledgements The authors wish to thank the NSERC for financial aid toward the support of Dr. N. Balakrishnan. Professor J. Srivastava, Past Editor-in-Chief, is thanked for her patient handling of this paper.

References Balakrishnan, N. and P.C. Joshi (1983). Single and product moments of order statistics from symmetricaUy truncated logistic distribution. Demonstratio Math. 16, 833-841. Balakrishnan, N. and P.C. Joshi (1984). Product moments of order statistics from doubly truncated exponential distribution. Naval Res. Logist. Quart. 31, 27-31. Barnett, V.D. (1966). Order statistics estimators of the location of the Cauchy distribution. J. Amer. Statist. Assoc. 61, 1205-1218 (Correction 63, 383-385). Birnbaum, A. and J. Dudman (1963). Logistic order statistics. Ann. Math. Statist. 34, 658-663. David, H.A. (1981). Order Statistics, 2nd edition. John Wiley & Sons, New York. Govindarajulu, Z. (1963). On moments of order statistics and quasi-ranges from normal populations. Ann. Math. Statist. 34, 633-651. Govindarajulu, Z. (1966). Best linear estimates under symmetric censoring of the parameters of a double exponential population. J. Amer. Statist. Assoc. 61,248-258. Gupta, S.S., A.S. Qureishi and B.K. Shah (1967). Best linear unbiased estimators of the parameters of the logistic distribution using order statistics. Technometrics 9, 43-56. Gupta, S.S. and B.K. Shah (1965). Exact moments and percentage points of the order statistics and the distribution of the range from the logistic distribution. Ann. Math. Statist. 36, 907-920. Joshi, P.C. (1978). Recurrence relations between moments of order statistics from exponential and truncated exponential distributions. Sankhya B 39, 362-371. Kjelsberg, M.O. (1962). Estimation of the parameters of the logistic distribution under truncation and censoring, Ph.D. Thesis, University of Minnesota. Shah, B.K. (1966). On the bivariate moments of order statistics from a logistic distribution. Ann. Math. Statist. 37, 1002-1010. Shah, B.K. (1970). Note on moments of a logistic order statistics. Ann. Math. Statist. 41, 2150-2152. Tarter, M.E. (1966). Exact moments and product moments of the order statistics from the truncated logistic distribution. J. Amer. Statist. Assoc. 61,514--525. Tarter, M.E. and V.A. Clark (1965). Properties of the median and other order statistics of logistic variates. Ann. Math. Statist. 36, 1779-1786.