Statistical Papers 42, 187-206 (2001) Statistical Papers �9 Spr inger -Ver lag 2001

Order statistics from non-identical right-truncated Lomax random variables with applications Aaron Childs l, N. Balakrishnan l, and M o h a m e d M o s h r e f z

1 Depar tment o f Ma thema t i c s and Statistics, McMaster Univers i ty , Hamil ton, Ontar io L8S 4K1, Canada

2 Depar tment o f Mathemat ics , A1-Azhar University, Nasr Ci ty 11884, Cairo, Egyp t

Received: N o v e m b e r 30, 1998; revised version: March 8, 2000

K e y W o r d s a n d P h r a s e s : Order statistics; outliers; robustness; single moments; prod- uct moments; recurrence relations; Lomax distribution; right-truncated Loraax distribution; permanents; censoring; bias; mean square error; BLUE.

A b s t r a c t : In this paper, we derive some recurrence relations for the single and the product moments of order stat ist ics from n independent and non-identically distributed Lomax and right-truncated Lomax random variables. These recurrence relations axe simple in nature and could be used systematical ly in order to compute all the single and product moments of all order statist ics in a simple recursive manner. The results for order statistics from the multiple-outlier model (with a slippage of p observations) axe deduced as special cases. We then apply these results by examining the robustness of censored BLUE's to the presence of multiple outliers.

1 I n t r o d u c t i o n

Let X h X 2 , ..., Xn be independent random variables having probabili ty density functions f l (x ) , f2(x) ...... fn (x ) and cumulative distribution functions F1 (x), F2(x), ...., Fn(x), respec- tively. Here, the Xi ' s axe independent and non-identically distributed (INID) random vari- ables. Let Xl:n < X2:n <_ ... < X,:n denote the corresponding order statistics obtained by arranging the n X~s in increasing order of magnitude. Then, the density function of Xr:n(1 < r _< n) can be writ ten as (David, 1981, p. 22).

188

1 r--1 f i fr:n(X) -- (r -- 1)!(7% -- r)! p ~ a__YI1 F i ' ( x ) f i r ( x ) {1 - Fib(X)},

= b = r + l

(1.1)

where ~-]~v denotes the summation over all n! permutations (il, i2, ...,in) of (1,2, ...,n). Sim- ilarly, the joint density function of Xr:n and Xs:n (1 < r < s < n) can be written as

1 r--1 s--1 f~ (~) Fi. (x) fi~ (x) fl (F,~(y) F~b (x) } fi,(Y) (,.- 1)~(s- , . - 1)~(n- s)~ ~ ]~ b=r-{-1

YI { 1 - T~(y)}, x < y. (1.2) c=8+I

Vaughan and Venables (1972) gave alternative forms of the densities in (1.1) and (1.2) in terms of permanents of matrices.

Balakrishnan (1994a,b) initiated work on recurrence relations for the single and product moments from INID random variables by considering the exponential, and right-truncated exponential distributions. Balakrishnan and Balasubramanian (1995) have done similar work for power function random variables. Childs and Balakrishnan (1998) considered the Pareto and truncated Pareto distributions, and Childs and Balakrishnan (2000) derived recurrence relations for the single moments of INID logistic random variables.

In this paper, we consider the case when the Xi 's are independent random variables having right-truncated Lomax distribution with density functions

f i ( X ) : f~i(1 -1- X) -(1-t-~') 1- (1 - I -T) -3* ' O < x < T , / 3 i > 0 (1.3)

and cumulative distribution functions

F i ( x ) = 1 - (1 + x ) - & 1 - ( I + T ) - O " 0 < x < T , 3 i > 0 (1.4)

for i ----- 1,2, . . . ,n. In the above equations, T denotes the point of truncation on the right of the Lomax distribution, and will be known in practical situtions as truncation is imple- mented by the experimenter.

It is clear from Eqs. (1.3) and (1.4) that the distributions satisfy the differential equa- tions

(1 + z)fi(z) = 3i{1 - Fi(z)} + Si

for i = 1, 2, ..., n where

(1.5)

189

~(1 § T) -f~' (1.6) S~= 1 - (1 + T)-~,"

If the X~'s are independent and identically distributed (~l = ~ . . . . . f~ = f~), and we let the truncation point T ~ co, then after introducing a scale parameter dr we obtain a random sample from the following distribution,

( / ( x )= ~- 1+ x>_O. (1.7) dr

This form of the Lomax distribution is also known as the Pareto distribution of the second kind; see for example Johnson, Kotz, and Balakrishnan (1994). Lomax (1954) used the distribution in (1.7) for the analysis of business failure data. Many of the applications of the Lomax distribution in (1.7) stem from the fact that it is a mixture (or compound) of e~(ponential distributions, f(xlo)(X[#) = Oe -~ where O has a gamma distribution, fo(0) = [dr/F(/~)](dr0)/~-le -a~ This is used, for example, in the analysis of heart transplant data in Turnbull et al. (1974), in the study of remission rates of psychiatric patients in Fox and Kraemer (1971), and in modelling the time intervals between industrial accidents in Maguire et al. (1952). See also Anscombe (1961), and Massy et al. (1970). A bivariate form of the Lomax distribution (1.7) has been used by Hutchinson (1979) to model the injuries to two drivers in a road accident, the severity of a lesion in a patient as assessed by two physicians, and as a model for the liability of two people who are related to each other of acquiring a certain disease.

In the case where X1,X2,...,X• are a random sample from (1.7), Malik (1966) has derived expressions for the means and covariances of the corresponding order statistics. Kulidorff and V ~ m m a n (1973) used these expressions to derive an exact explicit expression for the full sample best linear unbiased estimator (BLUE) of the scale parameter dr.

In this paper, Sections 2 and 3, we will make use of the differential equations in (1.5) to establish several recurrence relations satisfied by the single and the product moments of INID order statistics from (1.3). These relations will enable one to compute all the single and the product moments of all order statistics in a simple recursive manner. The results for the p-outlier model are then deduced as special cases in Section 4. In Section 5 we derive an expression for the BLUE of the scale parameter dr in (1.7) based on a Type-II right-censored sample, thus generalizing the work of Kulldorff and V~nnman (1973). We then apply our results from Section 4 by examining the robustness of the censored BLUE of dr in (1.7) to the presence of multiple outliers. Similar work for the normal distribution has been carried out by David and Shu (1978) for the single-outlier model. Their results are based on numerical computations of the corresponding moments of order statistics. The general results in this paper, however, allow us to consider the multiple outlier Ix)max model using exact calculations for the corresponding moments of order statistics.

190

2 Rela t ions for single moments

In this section, we present some recurrence relations satisfied by the single moments of order s tat is t ics by making use of the differential equat ions in (1.5). The proof of Resul t 2.1 is

given in the Appendix. . Let us denote the single moments E(Xrk:n) by p~(~) for 1 < r < n and k = 1,2, . . . , and the product moments E(Xr:n,Xs:n) by /zr, s:n for 1 < r < s < n.

Let us also use ' Bl(k) and ' [q to denote the single and the product moments of order I~r:n--1 /~r,s:n--I stat is t ics arising from n - 1 variables obtained by deleting Xi from the original n variables X1, X2, ..., Xn. Then, we have the following results. R e s u l t 2.1 for 2 < r < n - 1 and k = 0 ,1 ,2 , ....

n n 1)] ,,(k+l) = 1 ,-,., [i](k:+l) . . . . E ~ ' . ~ i - ( k ' { - 1) (k-{-l).(k:)Jc-~?.c(/3i-l-.Di).r_l:n_l--~_.lSi[J.~!(k__+ 1 , (2 .1)

1 1

where ~ / 3 i ~ k + 1. R e s u l t 2.2 For n >_ 2 and k = 0, 1,2, ...

•(k+l) ~_ 1 [ n n ]

nm ( • r ,6i) - (k + 1) (k + 1)p(~ ) -.1- ~ ( / 3 i -4- Si).[~]_(kl:'i-,nl.21 - - T k+l ~_4 Si , 1 1 "

where Y]? ~i ~ (k + 1). R e s u l t 2 .3 For k = 0,1, 2, ...,

(2.2)

p(k+l) 1 [ (k) S1Tk+l ] 1:1 ~31 -- (k + 1) (k + 1)/.z1:1

where 131 ~ (k + 1). R e s u l t 2 .4 For n _~ 2 and k = 0, 1, 2, ...

(2.3)

�9 (k+l) = 1 [ t ) ~ l : . ~- , ~ [q(k+l)] Wl:n ~-~? ~i - (k + 1) (k + "" (~) - 2_., oiU1:n-I ], (2.4)

1

where ~ ~ ~ (k + 1). R e m a r k 1: Resul ts 2.1-2.4 will enable one to compute all the single moments of all order s ta t is t ics in a simple recursive way for any specified values of 13i(i = 1 ,2 , . . , n ) and the t runca t ion point T. Recurrence relations for the unt runca ted case (T -* oo) are obtained by s imply set t ing S = 0 in the above results. R e m a r k 2: Alternatively, all the single moments of all order statistics could be computed in a s imple recursive manner using only Results 2.2 and 2.3, along with the following general

re la t ion established by Balakrishnan (1988),

191

where

3~r

:r = ~ ~ P~:S ( ' ) ISf=,

~ISI=, denotes the sum over al l subsets S of { 1 , 2 , . . . ,n } w i th cardinal i ty equal to r , and

h(k) is the k th moment of the i th order statistic among the Xj's, j E S i:S Remark 3: For the case when the Xi's are independent and identically distributed as right-truncated Lomax random variables (that is,/71 =/72 . . . . . /~n =/~), Results 2.1-2.4 reduce to

1 r . _ . . (k+l) ~ (k+1)1 i l i(rk:l) = n ~ -- T~ + 1)" [ (k + 1)~(r.k3 + n(~ + DJ/lr_l:n_ 1 -- nol~,.:,.,_lj,

w h e r e 2 < r < n , k > 0 , / 3 > 0 ,

[ ] = )P~-1: ,* -1 , n > 2, k >_ O, ~'n:n n /~ -- ( k + 1) (k + 1)p(~ ) + n(/3 + S" (k+l) _ n S T k + l

(2.6)

(2.7)

and

, (k+l) 1 [ ] - - I ) ~ I : I -- S ~ + 1 ( 2 . 8 ) ~'1:1 ~ - - (k '~- l ) ( k + ' " (k) 1

p(•+l) = 1 [ l : n n/~- ~ + 1) (k

w h e r e n j 3 ~ ( k + l ) , S = ~ .

~ (k) ,.-, (k+1)1 + 1}[ . t l : n - - n , .~ l :n_ l j (2.9)

3 Relat ions for product moments

In this section, we establish some recurrence relations satisfied by the product moments of order statistics using the differential equations in (1.5). The proof of Result 3.1 is given in the Appendix. Result 3.1 F o r n > 3 a n d 2 < r < n - 2 ,

192

1 [ - - + ~i)gr__ l,r:t~-- 1 /~ , .+I : . (E? /3 i ) - 2 /~:" + Pr + ~ ( 3 i [il

1

where )-~? ~i # 2. R e s u l t 3 .2

- L r ['1 l I " , t + . , ~ + t : . - i j , (3.1)

1[ Pt,2:2 -- )-~,2/3 i _ 2 #1:2 +P2:2 - T~_Sip .1 ,

1

where ~ fii # 2. R e s u l t 3 .3 For n _> 3,

1[ P-~-, . . . . = E ? f ~ - 2 g . - 1 : . + g . : .

where E~/~i # 2. R e s u l t 3 .4 For n _> 3,

n

T */~++n--2,n--l:n--I 1

T~-~S [~I ] - 2 _ , +/~-1:.-1J,

1

1 [ -r6+r l ]-1"1 . . . . ---- E T f ~ i - - 2 ~ t l : n -{ - /Xn :n / _ ~ ' M ' I : . - - l J ,

1

where ~ 3i # 2. R e s u l t 3 .5 For 2 < r < s < n - l a n d s - r > 2 ,

~ " : " = E ? ~ , - 2 [ ~ : " + g ' : " + ~ ( ~ ' - s " [,l ~ s ['] i -[- i ) ~ r - - l , s - - l : n - - 1 - - L i ~ r , s : n - - l J , I 1

where ~-~ A # 2. R e s u l t 3 .6 For 2 < r < n - 1,

1[ w .... = E ? Z , - 2 ~ : " + ~ : "

n r [i]

1

" 1 - r Es,.~!.-,, 1

where ~ ~i # 2.

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

193

R e s u l t 3 .7 For n > 3,

~tl,2:n = x--~ ~ ~l:n +/~2:~ - Si/~1,2:,-1 , Z . , 1 1

where E ~ / 3 / ~ 2. R e s u l t 3 .8 For n > 4 and 3 < s < n,

(3.7)

1 [ t,l ] # l , s : n ~'-~1 ~ i - - 2 1

(3.8)

where ~-:~ ~i ~ 2. R e m a r k 4: Resul ts 3.1-3.8 will enable one to compute all the product moments , and hence the covariances of all order statistics, in a simple recursive manner for any specified values of j3~ (i = 1, 2, ..., n) and the t runcat ion point T. Recurrence relations for the un t runca ted case (T --~ oo) are obta ined by simply setting S -- 0 in the above results. R e m a r k 5: Alternat ively, all of the product moments of all order statist ics could be computed in a simple recursive manner using only Results 3.1-3.3 and 3.7, along with the following general re lat ion due to Baiakrishnan, Bendre and Maiik (1992),

where

= _ �9 _ _ " - - - - i - - n .... ~j=n n~s+i+l ( 1)'+n r s+l(:_11) (Jn_81) (j)-~i,i+l: j, (3.9)

1

ISl~

~-~lSl=r denotes the sum over all subsets S of {1, 2 , . . . ,n} wi th cardinal i ty equal to r , and

~ d : s is the p roduc t moment of the i th and jth order statistic among the X j ' s , j E S R e m a r k 6: For the case when the Xi's are independent and identically d is t r ibuted as r ight- t runcated Lomax random variables (that is, ~1 =/32 . . . . . ]3n =/3), Resul ts 3.1-3.8 reduce to

1[ ] ~1,2:2 ~--- 2/3 - 2 ~1:2 + ]z2:2 - - 2TS~1:1 ; (3.10)

for n > 3

1[~1:n-l- ~2:n -- ~12:n-1]" (3.11)

194

f o r 2 < r < n - - 2

tgr , r+l :n

for n_> 3

//,n--l,n:n

f o r 3 < s < n - - 1

for n > 3

1[ : nf~ - 2 Pr:n + P~+l:n + n(]~ + S)p~-I . . . . . 1

-- n S ~ r , r + l : n - 1 ] ;

1[ - - n/3 - 2 ~ - l : n "Jr" ]An:n -q- n( f~ -I- S ) ~ - 2 , . - l : n - 1

- nSTp, n-l:n-1]

1[ ] ~l,s:n = n ~ - - 2 IZl:~ + I~s:n -- nS#l,s:n-1 ;

1[ ] ~ l ,n :n nf~ - 2 ~ l : n "~- ~ : n n S T l g l : n - 1

f o r 2 < r < s < _ n - 1 a n d s - r > _ 2

/~,*:n

f o r 2 < r < n - 2

~ l n : n - -

where n f l ~ 2 . In the above formulae,

1[ - - n]~ -- 2 P.r:n -f- ~s :n -}- n(]~ + S ) ] g . r - l , s - l : n - 1

- - nS~r,s:n-1] ;

1[ n~q~Tl~r:n-1]

S - - / ~ ( I + T ) -~ 1 - (1 + T)-f l

with T denoting the point of right truncation.

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

195

4 R e s u l t s for t h e m u l t i p l e - o u t l i e r m o d e l ( w i t h a s l i p p a g e o f p observations)

In this section, we consider the special case when X1,X2, ....,Xn-p are independen t right- t runca ted Lomax random variables with parameter /3 , while Xn-n+l, ..., Xn are independent r ight - t runcated Lomax random variables wi th parameter r (and independent of X1, X2, . . . , Xn-p. This s i tua t ion is known as mult iple-outl ier model with a s l ippage of p observations;

see Barne t t and Lewis (1994,pp. 66-68). Here, we denote the single moments by/~(~)[p] and the product moments by P~,s:,~[P].

In this s i tuat ion, the results established in sections 2 and 3 reduce to the following recurrence relations: (a) for k = 0, 1, 2,. . . ,

, (k+l) 1 [ ] -- 1)#1:1 [0] ; ,*a:l [0] / 3 - ( k + 1) ( k + (k) _ St~Tk+I

(b) for n >_ 2 and k = 0 ,1 ,2 , ...,

(4.1)

l : n I/~J 1 [ (k)

= ( n - p ) / 3 + p ' r - ( k + l ) ( k + l ) / ~ l : n [ p ]

- ~ s ' C k + X ) r~ 1]]; ( n - - p ) S / 3 , ~ . . k : _ l ~ [ p ] - It, v l . ~ l : n _ l t F - -

(c) for 2 < r < n - 1 and k = 0 ,1 ,2 , . . . ,

(4a)

p(k+X) ~:, [Pl 1 [ - S " (k+l) r ,

= ( n - - p ) / 3 + p T - - ( k + l ) (k+l)#(~)[P]+(n-P)(~3-v 0 ) # r - l : n - l W l

- - S " (k+x) ~ 1] ( n - -"S " ( k + l ) [-1 ~ ( k + l ) r 1] ] ; + p(T+ r ) /~r_l :n_l [p-- -- --p] Ot~,.:n_l~,l--P~rlar:n_l[ p - (4.3)

(d) for n k 2 and k = 0 ,1 ,2 , . . . ,

u~+l)[p] =

+

1 [ /r-n:nta"J ~O)]'t*n--l:n--li p] ( n - p ) / 3 + p ' r - ( k + l ) ( k + l ~ ' ( k ) r ' l + ( n - - P ) ( / 3 + S " (k+l) L,

, . _ ~ , (k+l) ] (4.4) P( + ~r)/~,~--X:n--l[P -- 1] -- ((n -p)S/3+pS.~)T k+l ;

(e)

= 2/312 (4.5)

196

(f) for n > 3,

(g) f o r 2 < r < n - 2 ,

= 1 [~l:n[P] "4- ~2:n[P] (n -- P ) 3 + p r -- 2

-- (n -- p)83/.tl,2:n_l[p] - p S r l Z l , 2 : n - l [ p - 1]]; (4.6)

~.r+~:. [p]

(h) for n > 3,

1 [ = (n - p)/3 + p ' r - 2 Pc:'*[P] + ]zr-l-l:n[P] q- (?2 - - p)(]~ -[- ~3)~$r-1 . . . . . 1 [P]

"1- p(T-t-,-,O~-)//.r--1 . . . . . l ip - 11 - ( n - - p ) S ~ l ~ - , r + l : n - l [ P ]

- - P S ~ P ~ , r + I : n - a [ P - 1]]; (4.7)

i [/tZn--l:.[p] -I- P-n:n[P] -I- (n -- p)(3-1- S , 3 ) # n - - 2 , . - - l : r t - - l [ P ] ~ - - I , . : . [ P ] = ('D. -- p ) ~ "~- pT -- 2

+ p ( r + S r ) l a n - 2 , n - l : n - l [ p - 1] - ( n - p ) S B T I . ~ - I : n - I [ P ]

- - p,.~rTld, n - l : n - l ~ P - 1]]; (4.8)

(i) f o r 3 < s < n - 1 ,

(j) for n > 3,

#l,,:.[p] 1 [

:" (n -- p)j3 -1- p v -- 2 #l:n[P]-l-/Zs:n[P]

-- ( n - - p ) S 3 p q . . . . . liP] --PS'TH.I,s:n-I[P-- 1]]; (4.9)

1 [ ~1 . . . . [P] = ( n - - p ) / ~ + p ' r - - 2 /U'l:~[P]-}-~tgn:v,[p]

-- ( n - - p ) S 3 T l z 1 : n - l [ p ] - p S ~ - T ~ I : n - I [ P - 1]];

(k) f o r 2 < r < s < n - l a n d s - r > 2 ,

(4.10)

1 [ = (n - p)/3 + p ' r - 2 P-r:n[P] -4=/Zs:n[P] + (n - p)(/~ + S/3)]-/.r-l,s-l:n-I [P]

+ p(~ + s ~ ) ~ - 1 , , - 1 : . - 1 [ p - 1] - (n - p ) S ~ . . . . . alp]

- p s r ~ . . . . . l i p - 1]]; (4.11)

197

(1) f o r 2 < r < n - 2 ,

. . . . N 1 [

= ( . - p ) Z + ~ - 2 v c : . [ p ] + ~ : . [ p ] + ( . - p ) ( Z + s , ) ~ - ~ , . _ ~ : . _ ~ [ p ]

+ p(~ + s . ) V ~ - l , . - l : . - l [ p - 1] - ( . -v)S~T~:.-1N

- p S r T p ~ : n - l ~ - 1]]. (4.12)

In the above formulae,

~ ( I + T ) - ~ a n d S r = T ( I + T ) - r (4.13) S Z = 1 - (1 + T ) - Z 1 - ( I + T ) -~

with T denoting the point of right truncation. Note that these recurrence relations reduce (by setting p -- 0 and j3 - 1) to those

presented earlier in Sections 2 and 3 for independent and identically distributed right- truncated Lomax random variables. Thus, by starting with the above equations for p ---- 0, all of the i.i.d, single and product moments can be determined. These same relations could then be used again, this time with p --- 1, to determine all the single and the product moments of all order statistics from a sample containing a single outlier. Continuing in this manner, the above relations could be used to compute all the single and the product moments (and hence covariances) of all order statistics from a p-outlier model in a simple recursive manner.

5 A p p l i c a t i o n s

Let

Xl:~ < x 2 : . _< . . . < xr :~ (5.1)

be a Type-II right-censored sample from the Lomax distribution in (1.7). Malik (1966) derived the following expressions for the expected value and covariance of order statistics.

1 E(Xi:n) = (Ai - 1)a if/3 > - - (i = 1, 2 , . . . , n)

n - i + l

C o v ( X i : n , X j : , ) = A j ( ~ .B~ - Ai)a 2

if j3 > m a x 1' (i <_ j ; i , j = 1 ,2 , . . . n) n - i + n - j + l

where

198

a n d

A i = F ( n - i q- 1 - l / 8 ) r ( n + 1) r ( n - i + 1 ) r ( n + 1 - 1 / 8 )

( i = l , 2 , . . . , n )

F ( n - i + 1 )F(n + 1 - 2 / 8 ) Bi = F ( n - i + 1 - 2 / 8 ) r ( n + 1) (i = 1, 2 , . . . , n)

B y mak ing use of the above expressions, and inver t ing the covar iance ma t r ix for the censored s a m p l e (5.1), as descr ibed for example in Arnold, Ba lak r i shnan , and Naga ra j a (1992), we have derived t he following exact explicit expression for the B L U E of a ,

where

1 [ r l ] ---- n 8 - 2 - D (8 § 1) ~ BiXi:n -b [8(n - r + 1) - 1]BrXr:n (5.2)

r - 1

D = (8 q- 1) ~ S i -b [~(n - r q- 1) - 1]Bn (5.3) i=1

Note t h a t th i s expression generalizes the full sample resul t g iven in Kulldorff and V~innman (1973).

B y mak ing use of the recurrence relat ions presen ted in Sect ion 4, we can now examine t he robus tnes s of t he censored B L U E of a to the presence of mul t ip le outl iers. As a n example, in Tab le 1 we present the bias and m e a n square error of the B L U E of a for n = 20, r -- 80, 90, a n d 100~0 of n, p ---- 1 , 2 , 3 , 8 ---- 5, and ~- --- 2 . 1 , 2 . 5 , 3 , 4 , 5 , 6 , 7 . Note t h a t t he outl iers occur w h e n 7- < 8; T ---- 8 is the I.I.D. case; and r > 8 is known as t he ' inl ier ' s i tuat ion.

F r o m Table 1 we see t h a t in the I.I.D. case (8 = r ---- 5), for each value of p the m e a n squa re error of t he censored B L U E increases only sl ightly as the level of censor ing increases. However, in t he t he most ex t reme out l ier s i tua t ion (~- = 2.1), t he m e a n square error of the uncenso red B L U E increases considerably, while t he MSE of t he censored B L U E suffers on ly a modes t increase. For example if p = 3 the MSE of the uncensored B L U E increases nea r ly five-fold f rom .0711 to .3468, while the censored B L U E based on 90% of the sample app rox ima te ly doubles f rom .0740 to .1456.

There fore we conclude t h a t if one suspects t h a t out l iers m a y be present in the sample, m a k i n g use of t he censored B L U E in (5.2) provides cons iderable p ro tec t ion a t a r a the r smal l p r e m i u m .

199

Acknowledgments

The first two authors would like to thank the Natural Sciences and Engineering Research Council of Canada for funding this research. This work was conducted during a summer visit of Dr. M. Moshref to McMaster University.

REFERENCES

Anscombe, F.J. (1961). Estimating a Mixed-Exponential Response Law, Journal of the American Statistical Association 56, 493-502.

Arnold, B.C., Balakrishnan, N., and Nagaraja, H.N. (1992). A First Course in Order Statistics, John Wiley & Sons, New York.

Balakrishnan, N. (1988). Recurrence relations for order statistics from n independent and non-identically distr ibuted random variables, Ann. Inst. Statist. Math. 40, 273-277.

Balakrishnan, N. (1994a). Order statistics from non-identical exponential random variables and some applications (with discussion), Comput. Statist. Data. Anal. 18, 203-253.

Balakrishnan, N. (1994b). On order statistics from non-identical right-truncated exponen- tial random variables and some applications, Uommum. Statist. - Theor. Meth. 23, 3373-3393.

Balakrishnan, N. and Balasubramanian (1995). Order statistics from non-identical power function random variables, Commum. Statist. - Theor. Meth. 24, 1443-1454.

Balakrishnan, N., Bendre, S.M., and Malik, H.J. (1992). General relations and identi- ties for order statist ics from non-independent non-identical variables, Annals of the Institute of Statistical Mathematics 44, 177-183.

Barnett, V. and Lewis, T. (1994). Oulters in Statistical Data, Third edition, John Wiley & Sons, Chichester.

Childs, A. and Balakrishnan, N. (1998). Generalized recurrence relations for moments of order statist ics from non-identical Pareto and truncated Pareto random variables with applications to robustness. In Handbook of Statistics - 16: Order Statistics: Theory & Methods (Eds., N. Balakrishnan and C.R. Rao), pp. 403-438, Elsevier Science, North-Holland, Amsterdam.

Childs, A. and Balakrishnan, N. (2000). Relations for single moments of order statist ics from non-identical logistic random variables and assessment of the effect of multiple outliers on the bias of linear estimators of location and scale, Submitted for Publication.

200

David, H.A. (1981). Order Statistics, Second edition, John Wiley & Sons, New York.

David, H. A., and Shu, V. S. (1978). Robustness of location estimators in the presence of an outlier, In Contributions to Survey Sampling and Applied Statistics: Papers in Honour of H. O. Hartley (Ed., H.A. David), 235-250, Academic Press, New York.

Fox, P.D. and Kraemer, H.C. (1971). A probability model for the remission rate of dis- charged psychiatric patients, Management Science 17B, 694-699.

Johnson, N.L., Kotz, S., and Balakrishnan, N. (1994). Continuous Univariate Distribu- tions - Volume 1, Second edition, John Wiley & Sons, New York.

Kulldorff, G., and V~mman, K., (1973). Estimation of the location and scale parameters of a Pareto distribution by linear fimctions of order statistics, Journal of the American Statistical Association 68, 218-227.

Maguire, B.A., Pearson, E.S. and Wynn, A.H.A. (1952). The time intervals between industrial accidents, Biometrika 39, 168-180.

Malik, H.J. (1966). Exact moments of order statistics from the Pareto Distribution, Skan- dinavisk Aktuarietidskrift, 49, No. 3-4, 144-157.

Massy, W.F., Montgomery, D.B. and Morrison, D.G. (1970). Stochastic Models of Buying Behavior, M.I.T. Press, Cambridge, Massachusetts.

Turnbull, B.W., Brown, B.W., Jr., and Hu, M. (1974). Survivorship analysis of heart transplant data, Journal of the American Statistical Association 69, 74-80.

Vaughan, R.J. and Venables, W.N. (1972). Permanent expressions for order statistics densities, J. Roy. Statist. Soc., Set. B 34, 308-310.

A p p e n d i x

P r o o f o f Resu l t 2.1: From (1.1), let us consider for 2 < r < n - 1, k -- 0, 1,2, ...

(r - 1)!(n - r~', L r''(k) . . . . + ~(~"+1)1 r - - 1

fi {I - F~b(X)} b=r+l

* 'T r - 1 n

a=l b=r

201

+ SirfoTXkHFio(X) {1 -- Fi~(x)}dx a = l b=r+l

upon using (1.5). Integrating now by parts treating x k for integration and the rest of integrand for differentiation, we get

(k + 1)(r - 1)!(n -- r)! [p(~) + ,'-r:n'O:+l)].[

r--1 r T r--1 n --'-- ~i .p -- ~ JO xk+l a=lH F'a ( x ) f ' , ( x ) b= r H { I - - Fib(X)}dx

a#j

fi( ] + II Fi~ (x) I - F~b(X)}dx j = r JO a = l s=r

r r--1 t T r--1 n

+ ~ Si l - X ]o xk+' I-I Fio(x)fi,(x) II { 1 - Fib(X)}dx L. j=l ~ b = r + l a:/:j

+ E fO x(k+l)H ~ ' ~ ~ fi {1- j j = r + l a = l b=r+l

b#j

(A.1)

Upon split t ing the first set of integrals (ones with negative sign) in the first term on the RHS of (A.1) into two each through the term 1 - Fir(x), we obtain

(k + 1)(r -- l)!(n -- r)! [p~(~ ) +

r - 1 T X k§ r = E ,.[Ef

p t . j= 1 JO a=l b=r+l a#j

s n + H Fi.(x)fi, (x) H { 1 - Fib(X)Idx j=r J O o~-1 b=~ b#j r--i T r--I f i ]

-- Ef 0 X k+l HFio(x)fij(x) (1 - F i b ( X ) } d x j = l a=l b = r + l a#j

+ S,. -i~=1 f _ go .=,l-IFi~ b=-,.+l{1 -- Fib(X))dx

a#j

202

~-~ f xk+l r-1 q- j=-~l Jo I I Fi.(x)fi#(x) {1 - F~b(x)}dx 6 = 1 b~r + l

b#i n

= (~Z~)(r - 1)!(n - ~)!,~.+') 1

v - - . [~)Ck+l) - (r - 2)!(r - 1)(n - r)! 2 ~pi#u,_l:n_ 1 1

~t

+ (~ - 1)!(. - ~ - ~)!(~ - ~) Z s,,~!~(~ +') 1

1

Result 2.1 is then derived simply by rewritng Eq. (A.2). Proof of Relation 3.1:

From (1.2), let us consider for 2 < r < n - 2

( r - 1 ) [ ( n - r - 1 ) ! [p~:n q- ffc,r+l:,~]

r--1

~ [ r [ r ( 1 + Y ) I I ~-(~)f'-(=)f' .+'(Y) l~I {1-F,~(y)}dydx "-~ p JO , I x a = l b----r@2

e T r - 1

a = l

where

(A.2)

(A.3)

x(~) = I T ( l + y ) : , . + , f y ) 1 ] { i -F , o (y ) )dy b~r--}-2

b = r ~ - I

+ s,.+, 1-I {1 - F,~(y)}dy b = r + 2

upon using (1.5). Integrating by parts now, we get

203

[ fi ~/: fi ] I(x) = Ji.+, - x {1 - F i , ( x ) } + y~,(y) {1 -Fi , (y)}dy b = r + l j=r+l b=r+l b#j

b=r+2 j = r + 2 z b=r+2 b#j

which, when subs t i tu ted in (A.3), gives

( r - 1)!(n - r - 1)! [/~r:n if- jtSr,r+l:n ]

= ~_.fli.+, - x2 n Fi.(x)fi.(x) {1 - Fib(x)}dx p a~l b=r+l

k/:/: -' h ] + xy H Fi.(xlfi.(xlfi,(y) {1 - Fi~(y)}dydx j=r+l a = l b=~+l

b~j

r~l h p a ~ l b=r+2

+ ~Y H F~-('lS,-('IS~,(Y) fI {1-F~A~)}@ j - - 2 a = l b=r+2

b # j

(A.4)

Next, from Eq. (1.2) let us write for 2 < r < n - 2

(r - 1)!(n -- r -- 1)! [Pr+l:n

where

q- P-r,r+ l:n]

-T -y r-- 1 f i

a=l b=r+2

b=r+2

(A.5)

204

r--1

J(y) = fo'(1 + x)fi~(x) 1-I Fi.(x)dx a ~ l

f y r - 1 = fl '"/o 1-[ F,~ - F,..(xlIdx

0 a = l ~ y r--1

+ Si. H Fi~ a = l

upon using (1.5). Integrati .rig by parts, we get

r - 1

r - 1 ~0y r - 1 -- ~ xf, j(x) H F , . ( x l { 1 - F , . ( x ) } d x

j = l afx apj

e t l r - - I 1

+ Jo = ~1 F~~ (=).f,. (=)d=J

--I- S,,. y Fi~ - xf,,(x) yI F,,,(x)d . j = l a=1

a#j

Upon subst i tut ing this expression of J(y) in (A.5), we get

(r - 1)!(n -- r - 1)! [ /~: , -J- I.tr,r-i.l:nl T r - 1 f i

= ~"~fl~"[fo Y2yIF~'(Y){1-F"(Y)}I"+'(Y) {1-F,~(yl}dy p a = l b=r+2

r - I t T t y r - I - ~/ / xyf,,(x) HF~o(X){1 -F'r(X)}S'.-l-l(~) H {I --F,b(Y)}dxdy

3=1 JO JO a=l b = r + 2 a# j

-T -y r - 1 ]

+ JO /0 xy H Fio(x)fi,.(x)fi,.+x(y) f i { 1 - Fib(y)}dxdy I a = l b = r + 2

r r T r--1 "1- ~_,.~ir[/O y2H Fio(y)fir+l(y ) f i {1-Fib(Y)}dy

p a = l b=r-l-2

r--1 r T r y r - 1 ]

- Jo_ Jo . j a-fj

205

(A.6)

On adding E~lS. (A.4) and (A.6) and simplifying the resulting expression, we obta in

(r-1)!(n-r-1)![pc:n + /*~+l:, +2P~,r+l:n}

1 n

i ) , . ~ v " ~,.~ - C,r - 9.~vc_,.,~ - l~c_,,n - ~ - 1 1 tz

- ( r - - 2 1 ! ( r - 1 ) ( n - r - 1 ) ! ~ - ' ~ [i} ~ i ] ' ~ r - - 1 , r : n - - 1 1

1 ~ K-" q . , [i] + (r - 1)!(n - r - 2)!(n - r - ,/_.., ~',t*,.r+l:n-1- 1

Result a.1 is derived s imply by rewriting the above equation.

206

Table 1. (Bias of Censored BLUE)Ia and (MSE)I(~ 2 In the Presence of Multiple Outliers

=This is the I.I.D. Case (p=O)