Mixture representation for order statistics from INID progressive censoring and its applications

Journal of Multivariate Analysis 99 (2008) 1999–2015 www.elsevier.com/locate/jmva

Mixture representation for order statistics from INID progressive censoring and its applications T. Fischer a , N. Balakrishnan b , E. Cramer a,∗ a Institute of Statistics, RWTH Aachen University, D-52056 Aachen, Germany b Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1

Received 11 April 2007 Available online 10 February 2008

Abstract In this paper, a mixture representation for the joint distribution function of progressively Type-II censored order statistics from heterogeneous distributions is established. Applications of this representation to stochastic orderings and inequalities are then illustrated. c 2008 Elsevier Inc. All rights reserved.

AMS 2000 subject classifications: 60E05; 60E15; 62N01 Keywords: INID progressive censoring; Multivariate stochastic order; Mixtures; Order statistics for independent nonidentically distributed variates; Permanents; Majorization

1. Introduction Progressively Type-II right censored order statistics based on heterogeneous (absolutely continuous) distributions were discussed by Balakrishnan and Cramer [8]. These authors developed the basic distribution theory for order statistics in this case. For example, they showed that the joint density of the progressively censored order statistics has a permanent expression, thus extending the corresponding well-known result for the usual order statistics due to [33]. Moreover, some explicit results were also derived when the distributions are exponential. In this paper, we obtain a mixture representation that relates the joint distribution function of the progressively Type-II censored order statistics to order statistics from independent and nonidentically distributed (INID) variables. This extends the corresponding result of [15] for the IID ∗ Corresponding author.

E-mail address: [email protected] (E. Cramer). c 2008 Elsevier Inc. All rights reserved. 0047-259X/$ - see front matter doi:10.1016/j.jmva.2008.02.007

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T. Fischer et al. / Journal of Multivariate Analysis 99 (2008) 1999–2015

case. However, the mixture form of Guilbaud [15] in the IID case involves weights which are not explicit and need to be computed in a recursive manner. He applied this property to establish nonparametric confidence bounds based on progressively Type-II censored order statistics. Further results on non-parametric inference are presented in [1]. At this point, it should be noted that extensions of IID progressive censoring have been developed in several other directions, too. For instance, a multivariate version has been proposed by Bairamov [3]. Random censoring schemes have been considered in, e.g., [34]. One may refer to the recent discussion paper by Balakrishnan [5] for an extensive review of all developments on progressive censoring. From the mixture representation for the INID case established here, we directly find explicit expressions for the weights in the mixture form of the joint distribution function of the progressively censored order statistics for the IID case. This leads to an explicit result for marginal distributions as well. The results here are in terms of joint distribution functions of some order statistics. Hence, there are no restrictions imposed either on the distributional assumptions or on the dependence structure. Furthermore, it is shown that results for the discrete progressively Type-II censored order statistics established recently by Balakrishnan and Dembinska [9] can be easily derived from this mixture result. Representations for the moments presented in [6] can also be easily seen from this mixture representation. Consequently, applications of this result to moments and (linear) recurrence relations are quite evident. Using the preservation property of the multivariate stochastic order under mixtures, we extend some stochastic order results obtained by Hu [16,17] (see also [24,19]) to INID progressively censored order statistics. Moreover, we obtain generalizations of the results of [29,30] for proportional hazard distributions. Finally, we present extensions of the inequalities for the distribution of the usual order statistics from INID variables established by Sen [31]. 2. Mixture representation Let X 1 , X 2 , . . . , X n , n ≥ 2, be independent random variables on a probability space (Ω , A, P) with distribution functions F1 , . . . , Fn , respectively, and m ∈ N = {1, 2, 3, . . .}, m ≤ n. Moreover, denote by Nm 0 the m-fold cartesian product of N0 = N ∪ {0}. Progressively Type-II censored order statistics from X 1 , X 2 , . . . , X n can be constructed as follows: Suppose that X 1 , X 2 , . . . , X n are random lifetimes corresponding to n independent units R on a life-testing experiment. At the first failure time, i.e., min{X 1 , X 2 , . . . , X n } = X 1:m:n , a prefixed number R1 of surviving units are removed at random from the sample. Then, at the time of R the first failure of the remaining n − 1 − R1 units, denoted by X 2:m:n , a pre-fixed number R2 of R surviving units are randomly removed, and so on. Finally, at the mth failure time X m:m:n , all the remaining surviving Rm = n − m − R1 − R2 − · · · − Rm−1 units are censored. Pm Given a progressive censoring scheme R = (R1 , R2 , . . . , Rm )0 ∈ Nm i=1 Ri = 0 with m + Pm R (R + 1) = n, the random variable X is then the ith progressively Type-II censored i i=1 i:m:n order statistic, i ∈ {1, 2, . . . , m}, and R R R (X 1:m:n , X 2:m:n , . . . , X m:m:n )0

is the vector of the progressively Type-II censored order statistics. Note that when m = n and Ri = 0 for every i ∈ {1, 2, . . . , m}, we simply have R R R (X 1:n:n , X 2:n:n , . . . , X n:n:n )0 ≡ (X 1:n , X 2:n , . . . , X n:n )0 ,

where (X 1:n , X 2:n , . . . , X n:n )0 is the vector of the usual order statistics.

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For ω ∈ Ω , the construction of a progressively Type-II censored sample 0  R R R X 1:m:n (ω), X 2:m:n (ω), . . . , X m:m:n (ω) can be done by the following procedure. Construction 2.1 (Generation of Progressively Censored Order Statistics). Using the notation introduced above, the progressively Type-II-censored order statistics R R R (X 1:m:n , X 2:m:n , . . . , X m:m:n )0

from a sample X 1 , . . . , X n can be constructed as follows. Let ω ∈ Ω be arbitrary. Then: Determine the order statistics (X 1:n (ω), X 2:n (ω), . . . , X n:n (ω))0 ; Set N = {1, 2, . . . , n} and i = 1; R Let ki = min(N ) and define X i:m:n (ω) = X ki :n (ω); Choose a without-replacement sample Ri ⊆ N \ {ki } with |Ri | = Ri at random (from N \ {ki }); Step 5. Set N = N \ [{ki } ∪ Ri ] and i = i + 1; Step 6. If i ≤ m, then go to Step 3, else stop. Step Step Step Step

1. 2. 3. 4.

Evidently, the vector of the progressively Type-II censored order statistics is then given by  0
0 R R R X 1:m:n (ω), X 2:m:n (ω), . . . , X m:m:n (ω) = X k1 :n (ω), X k2 :n (ω), . . . , X km :n (ω) . It is important to mention here that the above construction of progressively Type-II censored order statistics does not impose any assumptions either regarding the distributions or the dependence structure on the variables X 1 , . . . , X n . This facilitates a probabilistic study of the progressively Type-II censored order statistics in the INID case. We assume that the choice of the sets R1 , R2 , . . . , Rm is modeled by a random variable Z , where Z has a discrete uniform distribution and is stochastically independent of X 1 , X 2 , . . . , X n . It is worth mentioning that this assumption may be relaxed. For instance, Ng and Chan [28] propose a model where the removal probabilities are sequentially chosen. In particular, their model assumes that these probabilities depend on the observed failure times. Then, the following theorem presents a mixture representation for the joint distribution of the progressively Type-II censored order statistics in this general INID set-up. Theorem 2.2 , . . . , R m )0 ∈ N m 0 Pm(Mixture Representation for the INID Case). Let R = (R1P m with m + i=1 Ri = n and x1 , x2 , . . . , xm ∈ R. Further, let γ j = i= j (Ri + 1) = P j−1 n − ( j − 1) − i=1 Ri . We then have R R R P(X 1:m:n ≤ x1 , X 2:m:n ≤ x2 , . . . , X m:m:n ≤ xm ) X 1 P(X k1 :n ≤ x1 , X k2 :n ≤ x2 , . . . , X km :n ≤ xm ), = m  Q γi −1  R i=1

Ri

Z

where the summation is over all elements (k1 , R1 , . . . , km , Rm ) of the set

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( (l1 , S1 , l2 , S2 , . . . , lm , Sm ) : l1 = 1, S1 ⊆ {2, . . . , n}, |S1 | = R1 ,

ZR =

" li = min {1, . . . , n} \

#!

i−1 [

(S j ∪ {l j })

,

j=1

" Si ⊆ {1, . . . , n} \

i−1 [ j=1

Sj ∪

#

i [

)

{l j } , |Si | = Ri ∀i = 2, . . . , m .

(2.1)

j=1

Proof. Let Z model the choice of the sets R1 , . . . , Rm in Construction 2.1. According to our assumptions, Z is a random variable with values in Z R , which has a discrete uniform distribution and is independent of X 1 , . . . , X n . Consequently, we may assume without loss of any generality that Z is defined on the same probability space as X 1 , XP 2 , . . . , X n , for otherwise we can consider an appropriate product space. Since γi = n − i + 1 − i−1 j=1 R j denotes the number of units in the experiment before the ith failure, i.e., γi is the cardinality of the set N in the algorithm before the ith run, a simple combinatorial argument readily yields  m  Y γi − 1 |Z | = . Ri i=1 R

Hence, it follows that R R R P(X 1:m:n ≤ x1 , X 2:m:n ≤ x2 , . . . , X m:m:n ≤ xm )  X  R R = P X 1:m:n ≤ x1 , . . . , X m:m:n ≤ xm |Z = (k1 , R1 , . . . , km , Rm )

ZR

× P (Z = (k1 , R1 , . . . , km , Rm ))  X  R R = P X 1:m:n ≤ x1 , . . . , X m:m:n ≤ xm |Z = (k1 , R1 , . . . , km , Rm ) ZR

#−1 m  Y γi − 1 × Ri i=1 X 1 = m  P(X k1 :n ≤ x1 , X k2 :n ≤ x2 , . . . , X km :n ≤ xm ),  Q γi −1 R "

i=1

Ri

Z

where the summation is over Z R defined in Eq. (2.1).

C

By the uniqueness theorem for probability measures, the joint distribution of the progressively Type-II censored order statistics is therefore a convex combination of distributions of vectors of the usual order statistics. Remark 2.3. It should be noted that the number k1 in the summation in the above mixture representation is fixed and equals 1. From Construction 2.1, we then obtain the following bounds for k2 , . . . , km :

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2 ≤ k2 ≤ R1 + 2, k2 + 1 ≤ k3 ≤ k2 + R1 + R2 + 1, .. . m−1 X km−1 + 1 ≤ km ≤ km−1 + R j + 1 = km−1 + n − Rm − m + 1. j=1

Since γm = Rm + 1, we also have  m−1 m  Y  γi − 1  Y γi − 1 = Ri Ri i=1 i=1 reflecting simply that in the last step of the construction Rm out of the remaining Rm units are removed. Remark 2.4. The proof of Theorem 2.2 does not rely on the dependence structure of the sample X 1 , . . . , X n . Therefore, the mixture representation holds for an arbitrary dependence structure. The same holds for the distributional assumptions as well. Corollary 2.5.PLet Bm denote the Borel σ -field on Rm . Moreover, let R = (R1 , R2 , . . . , Rm )0 ∈ m Nm i=1 Ri = n and γi , i ∈ {1, 2, . . . , m}, be defined as in Theorem 2.2. Then, for all 0 with m + Borel sets B ∈ Bm :   R R R P (X 1:m:n , X 2:m:n , . . . , X m:m:n )0 ∈ B X
1 = P (X k1 :n , X k2 :n , . . . , X km :n )0 ∈ B ,   m−1 Q i=1

γi −1 Ri

ZR

where the summation is over Z R defined in Eq. (2.1). Remark 2.6. Let ∅ 6= I $ {1, 2, . . . , m}. Then, by considering R R R lim P(X 1:m:n ≤ x1 , X 2:m:n ≤ x2 , . . . , X m:m:n ≤ xm )

x j →∞ j6∈ I

from the mixture representation in Theorem 2.2,
we obtain a corresponding mixture R representation for the marginal distribution of X i:m:n . This yields, for example, expressions i∈I for density functions, moments, etc. In the case of INID random variables X 1 , . . . , X n , the joint and marginal density functions as well as the one-dimensional distribution and survival functions of usual order statistics can be expressed as permanents; see, for example, [33,2], and the recent review article by Balakrishnan [4]. The densities and distribution functions of progressively Type-II censored order statistics can therefore be written as mixtures of permanents. Though Balakrishnan and Cramer [8] showed that the joint density function has a permanent representation, they did not obtain result for marginal distributions. Theorem 2.2, however, presents explicit expressions for both distribution and density functions as mixtures of permanents for joint as well as marginal distributions of progressively Type-II censored order statistics. In the IID case, we obtain the following mixture representation.

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Corollary 2.7 (Mixture Representation for the IID Case). Suppose that XP 1 , . . . , X n are IID m random variables with cdf F. Let R = (R1 , . . . , Rm )0 ∈ Nm with m + i=1 Ri = n and 0 m γi , i ∈ {1, . . . , m}, be as in Theorem 2.2. Then, for all Borel sets B ∈ B  m  Q Rm−1∗ −i m−2∗ R1 R2∗ −i 1∗  X  X X R R R P (X 1:m:n , X 2:m:n , . . . , X m:m:n )0 ∈ B = ··· i 1 =0 i 2 =0

i m−1 =0

j=2

m−1 Q  j=1

× P (X 1:n , X i1∗ +2:n , . . . , X im−1∗ +m:n ) ∈ B , Pj Pj where R j∗ = ν=1 Rν , 1 ≤ j ≤ m, i j∗ = ν=1 i ν , 1 ≤ j ≤ m − 1. 0




R j∗ −i j−1∗ Rj γ j −1 Rj

 (2.2)

Proof. First, we rewrite the indices of the order statistics in (2.2) in terms of the differences i j = k j+1 − k j − 1, 1 ≤ j ≤ m − 1, i m = n − km , k1 = 1. After some simple calculations, we find k j = i 1 + · · · + i j−1 + j = i j−1∗ + j,

1 ≤ j ≤ m,

where i 0∗ = 0. Since the underlying random variables have the same distribution function F, the distribution of (X 1:n , X i1∗ +2:n , . . . , X im−1∗ +m:n )0 depends only on i 1∗ , . . . , i m−1∗ and not on the censoring scheme (R1 , . . . , Rm ). Initially, these numbers are supposed to be fixed. Thus, we have to count the elements of Z R (see Eq. (2.1)) leading to this particular random vector. This means that we have to figure out how many elements (1, R1 , i 1∗ + 2, R2 , . . . , i m−1∗ + m, Rm ) exist. However, the set Z R is not appropriate for this purpose since the order of the elements in the sets Ri is not taken into account. So, we introduce the set e Z R , where the order of the elements in R1 , . . . , Rm is accounted for. Clearly, the identity ! ! m m m−1 Y Y Y γj − 1 R R e |Z | = (2.3) R j ! |Z | = Rj! Rj j=1 j=1 j=1 holds. Due to Construction 2.1, we have the following restrictions (for fixed ω ∈ Ω ): The order statistics X 2:n (ω), . . . , X i1∗ +1:n (ω) must be censored in the first step, i.e., {2, . . . , i 1∗ + 1} is a subset of the R1 indices randomly chosen in the first step. The order statistics X i1∗ +3:n (ω), . . . , X i2∗ +2:n (ω) have to be censored either in the first or in the second step of the progressive censoring procedure. Thus, the corresponding indices have to be chosen out of R1 − i 1 remaining numbers from the first censoring step and from R2 indices of the second censoring step. In general, it turns out that in the jth step i j random variables have to be chosen out of R j∗ − i j−1∗ (taking into account the order of removal). The total number of possibilities in the jth step is thus   (R j∗ − i j−1∗ )! R j∗ − i j−1∗ ij! = . ij (R j∗ − i j∗ )! For j = m, this expression simplifies to i m ! since Rm∗ −i m−1∗ = i m . Noting that R1∗ −i 0∗ = R1 , we have  m m m  Y Y Y (R j∗ − i j−1∗ )! R j∗ − i j−1∗ = Rj! . (2.4) (R j∗ − i j∗ )! Rj j=1 j=1 j=2 Upon combining (2.3) and (2.4), we arrive at the coefficients in (2.2).

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Finally, note that i 1 , . . . , i m−1 have to be chosen according to i 1 ∈ {0, . . . , R1 }, i 2 ∈ {0, . . . , R2 + R1 − i 1 } = {0, . . . , R2∗ − i 1∗ }, .. . i m−1 ∈ {0, . . . , Rm−1∗ − i m−2∗ }. This establishes the mixture representation in (2.2).

(2.5)

C

Remark 2.8. Corollary 2.7 can be proved in an alternate way as follows. Fix 1 = k1 < · · · < km ≤ n − Rm . We now consider Construction 2.1 backwards. In the last step, Rm units are censored subject to the restriction that their numbers are larger than km . This results in     Rm∗ − i m−1∗ n − km = Rm Rm choices. In the preceding step, Rm−1 units have to be censored such that their indices are larger than km−1 and do not equal km . Moreover, they are not censored in the final step. This results in     n − km−1 − 1 − Rm Rm−1∗ − i m−2∗ = Rm−1 Rm−1 possibilities. In the jth step, using a similar argument, we arrive at the binomial coefficient     R j∗ − i j−1∗ n − k j − (m − j) − Rm − · · · − R j+1 . = Rj Rj   In the first step, we have RR11 = 1. Combining all these possibilities, we arrive at the expression in (2.4). Remark 2.9. The mixture representation established in Corollary 2.7 can also be obtained directly using the joint density function of a progressively Type-II censored sample R R R X 1:m:n , X 2:m:n , . . . , X m:m:n , i.e., fXR

R R 1:m:n ,X 2:m:n ,...,X m:m:n

(x1 , . . . , xm ) = c

m Y

f (x j ){1 − F(x j )} R j ,

x1 ≤ · · · ≤ xm ,

(2.6)

j=1

Q where c = mj=1 γ j (see, e.g., [6]). First, we illustrate the method when m = 3. The proof is based on the binomial theorem using which we first express {1 − F(x1 )} R1 = {[1 − F(x2 )] + [F(x2 ) − F(x1 )]} R1  R1  X R1 {F(x2 ) − F(x1 )}i1 {1 − F(x2 )} R1 −i1 . = i 1 i =0

(2.7)

1

Then, collecting the powers of 1 − F(x2 ) and applying the binomial theorem again, we have {1 − F(x2 )} R2 +R1 −i1 R1 +R X2 −i1  R1 + R2 − i 1  = {F(x3 ) − F(x2 )}i2 {1 − F(x3 )} R1 +R2 −i2∗ . i 2 i =0 2

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Upon combining these results, we obtain the expression

fXR

R R 1:3:n ,X 2:3:n ,X 3:3:n

(x1 , x2 , x3 ) = c

3 Y

f (x j )

  R1 R2∗ −i 1∗  X X R2∗ − i 1∗ R1 i 1 =0 i 2 =0

j=1

i2

i1

× {F(x2 ) − F(x1 )}i1 {F(x3 ) − F(x2 )}i2 {1 − F(x3 )} R3∗ −i2∗   R1 R2∗ −i 1∗  X X R2∗ − i 1∗ i 1 !i 2 !(R3∗ − i 2∗ )! R1 =c g1,i1∗ +2,i2∗ +3:n (x1 , x2 , x3 ), i2 i1 n! i =0 i =0 2

1

where g1,i1∗ +2,i2∗ +3:n denotes the joint density of the usual order statistics X 1:n , X i1∗ +2:n , X i2∗ +3:n ; see [14]. Noticing that γ1 = n, γ3 − 1 = R3 , γ j+1 = γ j − R j − 1, j = 1, 2, and  c

R1 i1

=

=



R2∗ − i 1∗ i2



3 Y R1 !(R2∗ − i 1∗ )!(R3∗ − i 2∗ )! i 1 !i 2 !(R3∗ − i 2∗ )! = γj n! (R1 − i 1 )!(R2∗ − i 2∗ )!n! j=1

R1 !(R2∗ − i 1∗ )!(R3∗ − i 2∗ )!R2 !R3 !γ2 !γ3 ! (R2∗ − i 1∗ − R2 )!R2 !(R3∗ − i 2∗ − R3 )!R3 !(γ1 − 1)!(γ2 − 1)!(γ3 − 1)!    R2∗ −i 1∗ R2



γ1 −1 R1

R3∗ −i 2∗ R3



γ2 −1 R2



.

For m = 3, this proves that the mixture representation (2.2) results. Now, the extension to m progressively censored order statistics follows by induction. Consider the joint density in (2.6) with m ≥ 2. Assume that the mixture representation holds for any sample of first m − 1 progressively censored order statistics. Using the expansion (2.7), we find

fXR

R R 1:m:n ,X 2:m:n ,...,X m:m:n

= f (x1 )

F h i1 (x2 , . . . , xm )

i1

=

m Y

f (x j ){1 − F(x j )} R j ,

j=1

 R1  X R1 i 1 =0

where

(x1 , . . . , xm ) = c

{F(x2 ) − F(x1 )}i1

Qm−1 j=1



F γj

γ1 γ2 F h (x2 , . . . , xm ) γ1 − 1 − i 1 i 1

f (x j+1 ){1 − F(x j+1 )}

F

Rj

 denotes the joint density of F

F

the first m − 1 progressively censored order statistics with censoring scheme (R1 , . . . , Rm−1 ). F

F

This scheme is given by R1 = R1 + R2 − i 1 , R j = R j+1 , 2 ≤ j ≤ m − 1. Notice that F

F

γ1 = γ2 + R1 − i 1 = γ1 − 1 − i 1 and that γ j = γ j+1 , 2 ≤ j ≤ m − 1. Hence, we can apply F

F

F

the induction assumption on the joint densities h i1 , 0 ≤ i 1 ≤ R1 , so that with i 0∗ = 0, i 1∗ = i 2 , F

F

i j∗ = i j−1∗ + i j+1 , 2 ≤ j ≤ m,

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fXR

R R 1:m:n ,X 2:m:n ,...,X m:m:n

F R1

×

X

···

i 2 =0 i 3 =0

i1

i 1 =0

F F Rm−2∗ −i m−3∗

F F R2∗ −i 1∗

X X

(x1 , . . . , xm ) = f (x1 )

 R1  X R1

m−1 Q



F

F

R j∗ −i j−1∗

{F(x2 ) − F(x1 )}i1

γ1 γ2 γ1 − 1 − i 1



F

Rj

j=2

  g1,i F +2,...,i F +m−1:γ F (x2 , . . . , xm ). m−2 1∗ m−2∗ 1 Q γ jF −1

i m−1 =0

F

j=1

Rj

(2.8) A closer look at g1,i F +2,...,i F

F m−2∗ +m−1:γ1

1∗

yields the identity

f (x1 ){F(x2 ) − F(x1 )}i1 g1,i F +2,...,i F

F m−2∗ +m−1:γ1

1∗

=

(x2 , . . . , xm )

(γ1 − 1 − i 1 )!i 1 ! g1,i1 +2,...,im−1∗ +m:n (x1 , . . . , xm ). γ1 ! F

F

Observing that R j∗ − i j−1∗ = R j+1∗ − i j∗ , 1 ≤ j ≤ m − 1, Eq. (2.8) reads fXR

R R 1:m:n ,X 2:m:n ,...,X m:m:n

(x1 , . . . , xm )  m  Q R j∗ −i j−1∗

=

R1 R2∗ −i 1∗ X X i 1 =0 i 2 =0

···

Rm−1∗ −i m−2∗ X i m−1 =0

j=3 m−1 Q  j=3

Rj γ j −1 Rj



R1 !γ2 (R2∗ − i 1∗ )! (γ2 − 1 − R2 )! (R1 − i 1 )! (γ1 − 1)!

× g1,i1 +2,...,im−1∗ +m:n (x1 , . . . , xm ). Since 

R2∗ −i 1∗ R2



R1 !γ2 (R2∗ − i 1∗ )! (γ2 − 1 − R2 )!   = γ1 −1 γ2 −1 (R1 − i 1 )! (γ1 − 1)! R1

R2

we arrive at the desired expression. In particular, the above derivation illustrates how the joint density of progressively censored order statistics given in (2.6) can be derived from the mixture representation. In fact, these calculations along with the mixture representation give a formal proof of the joint density in the IID case. Remark 2.10. Guilbaud [15] presented a similar result for progressively Type-II censored order statistics from an IID sample X 1 , . . . , X n . He expressed the ith progressively Type-II R censored order statistic X i:m:n , i ∈ {1, . . . , m}, as a scalar product of the vector of order R,...,w R )0 , which is independent of 0 statistics (X 1:n , . . . , X n:n ) and a random vector (wˆ i1 ˆ in 0 (X 1:n , . . . , X n:n ) and has a multinomial distribution, as follows: R X i:m:n =

n X

wˆ iRj X j:n .

j=1

Through this representation, Guilbaud [15] derived confidence intervals for the p-quantile (0 < p < 1) of the common distribution of X i ’s.

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This representation can, in fact, be derived from Corollary 2.7 by calculating the onedimensional marginal distributions. Our approach, however, leads to explicit expressions for the weights, i.e., the probabilities of the multinomial distribution in the formulation of Guilbaud [15] who presented only an iterative procedure to compute these probabilities. Moreover, our results also hold for the random vector of progressively Type-II censored order statistics. In particular, we obtain for B ∈ B1 R P(X m:m:n ∈ B) =

n−R Xm ν=m

αν P(X ν:n ∈ B),

where the explicit weights αν ’s are given by m  Q αν =

R1 R2∗ −i 1∗ X X

···

Rm−1∗ −i m−2∗ X

i 1 =0 i 2 =0 i m−1 =0 i 1 +···+i m−1 =ν−m

j=2

(2.9)

R j∗ −i j−1∗ Rj

m−1 Q  j=1

γ j −1 Rj



 ,

m ≤ ν ≤ n − Rm .

Note that, for 1 ≤ j ≤ m, a progressively censored order statistic X Rj:m:n can always be seen as a (maximum) progressively censored order statistic with an appropriately chosen censoring Pm scheme, i.e., X Rj:m:n ∼ X Sj: j:n with S = (R1 , . . . , R j−1 , S j )0 and S j = i= (R i + 1) − 1. j Formula (2.9) provides an approach to calculate the distribution of a progressively censored order statistic. Alternatively, the Kamps–Cramer representation of the distribution function (see [18]) ! j j j Y X 1 Y (γν − γi )−1 (1 − F(t))γi , t ∈ R, FX j:m:n (t) = γi f (t) γ i ν=1,ν6=i i=1 i=1 can be considered. Though the representation in (2.9) is more complicated than the Kamps–Cramer representation, which is the one preferable for computational purposes, it is certainly of interest and use for theoretical reasons as developments in subsequent sections demonstrate, for example. Remark 2.11. It is important to observe that Theorem 2.2 and Corollary 2.7 do not impose restrictions such as absolute continuity on the distribution functions F1 , . . . , Fn . Hence, these results can also be applied to discrete distributions. This yields in the IID case the quantile representation of the progressively Type-II censored order statistics obtained recently by Balakrishnan and Dembinska [9], i.e., R (X Rj:m:n )1≤ j≤m ∼ (F −1 (U j:m:n ))1≤ j≤m , R where U j:m:n , 1 ≤ j ≤ m, are progressively Type-II censored order statistics based on the uniform distribution. Moreover, the distribution theory can be easily developed using the mixture representation. For example, Theorem 1 of [9] follows by starting with the well-known expression [see [27,20]] Z P(X k1 :n = x1 , . . . , X km :n = xm ) = fUk1 :n ,...,Ukm :n (u 1 , . . . , u m )du 1 . . . du m , A

where A = [F(x1 −), F(x1 )] × · · · × [F(xm −), F(xm )]

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and Uk1 :n , . . . , Ukm :n are uniform order statistics, and using the mixture representation and the linearity of the integral. Other distributional results such as marginal distributions and moments can be obtained in a similar manner. 3. Stochastic ordering of progressively Type-II censored order statistics In this section, we use the mixture representation in Theorem 2.2 in order to extend some stochastic ordering results for the usual order statistics to the case of progressively Type-IIcensored order statistics. The results are based on the following general assumptions. Let n ∈ N, n ≥ 2, m ∈ N, m ≤ n, (Ω , A, P) be a probability space, and X 1 , . . . , X n and Y1 , . . . , Yn be independent random variables on (Ω , A, P). X i and Yi are assumed to have absolutely continuous distribution functions Fi and G i , respectively, 1 ≤ i ≤ n. Moreover, let X 1:n , . . . , X n:n and Y1:n , . . . , Yn:n , respectively, denote the corresponding order Pm statistics. For a given censoring scheme R = (R1 , . . . , Rm )0 ∈ Nm with m + R = n, i i=1 0 R R R R , . . . , X m:m:n and Y1:m:n , . . . , Ym:m:n denote the corresponding progressively Type-II let X 1:m:n censored order statistics based on the X - and Y -samples, respectively. Finally, for two vectors x = (x1 , . . . , xn )0 , y = (y1 , . . . , yn )0 ∈ Rn , we define x ≤ y ⇐⇒ xi ≤ yi

∀i ∈ {1, 2, . . . , n}.

The first generalization is regarding the multivariate stochastic order of vectors of order statistics. Definition 3.1. We have the following definitions: 1. Let X and Y be arbitrary random variables. Then, X is said to be larger than Y in the stochastic order, denoted by X ≥st Y , if P(X > t) ≥ P(Y > t) for all t ∈ R. 2. A set U ⊆ Rn is called an upper set if y ∈ U whenever y ≥ x and x ∈ U . 3. Let X = (X 1 , . . . , X n )0 and Y = (Y1 , . . . , Yn )0 be arbitrary random vectors. Then, X is said to be larger than Y in the multivariate stochastic order, denoted by X ≥st Y , if P(X ∈ U ) ≥ P(Y ∈ U ) for all Borel-measurable upper sets U ⊆ Rn . The following definitions regarding majorization are from [26]. Definition 3.2. Let λ = (λ1 , . . . , λn )0 , µ = (µ1 , . . . , µn )0 ∈ Rn , and λ[1] ≥ · · · ≥ λ[n] , µ[1] ≥ · · · ≥ µ[n] be their ordered components. Then, the vector µ is said to be majorized by the vector λ, denoted by λ  µ, if m X

λ[i] ≥

m X

µ[i] ,

m = 1, 2, . . . , n − 1,

n X

λ[i] =

i=1

i=1

i=1

and

n X

µ[i] .

i=1

µ is said to be weakly supermajorized by λ, denoted by λ w µ, if n X i=m

λ[i] ≤

n X

µ[i] ,

m = 1, 2, . . . , n.

i=m

Obviously majorization implies weak supermajorization. The following result is then obtained from Theorem 2.2 and its proof follows directly from Corollary 2.5 and the facts that the multivariate stochastic order is closed under marginalization and mixtures; see, for example, [32, Theorem 6.B.16, p. 273].

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Theorem 3.3. Let X () = (X 1:n , . . . , X n:n )0 and Y() = (Y1:n , . . . , Yn:n )0 . Then, for an arbitrary censoring scheme R, we have R R R R X () ≥st Y() H⇒ (X 1:m:n , . . . , X m:m:n )0 ≥st (Y1:m:n , . . . , Ym:m:n )0 .

We can now exploit Theorem 3.3 to transform a result of [16,17] for the usual order statistics to the case of progressively Type-II censored order statistics. Theorem 3.4. Let X 1 , . . . , X n and Y1 , . . . , Yn be independent non-negative random variables with survival functions F i (t) = e−H (λi t) ,

G i (t) = e−H (µi t) ,

t ≥ 0, 1 ≤ i ≤ n, Rt

where λ1 , . . . , λn , µ1 , . . . , µn > 0 and H (t) = 0 h(s)ds, t ≥ 0, with a non-negative integrable function h such that limt→∞ H (t) = ∞. Suppose h is decreasing and the function h ∗ defined by h ∗ (x) = xh(x) is increasing. Then, (λ1 , . . . , λn )0 w (µ1 , . . . , µn )0 implies R R R R R R (X 1:m:n , X 2:m:n , . . . , X m:m:n )0 ≥st (Y1:m:n , Y2:m:n , . . . , Ym:m:n )0 .

The preceding result can be readily applied to some parametric families of distributions. Hu [16] considered, for example, Weibull, gamma and half normal distributions. In particular, he established for the family α

F(t) = 1 − e−t ,

t ≥ 0, 1 ≤ i ≤ n,

with shape parameter α > 0, the following result. Theorem 3.5. For α ∈ (0, 1] and λ = (λ1 , . . . , λn )0 , µ = (µ1 , . . . , µn )0 ∈ (0, ∞)n with λ w µ, let X i ∼ Weibull(α, λi ) and Yi ∼ Weibull(α, µi ), i ∈ {1, 2, . . . , n}. Then, (X 1:n , X 2:n , . . . , X n:n )0 ≥st (Y1:n , Y2:n , . . . , Yn:n )0 . This result with λ  µ was obtained later by [19]; see [32, p. 319 and Example 6.B.27]. For gamma distributions, Hu [16] proved the following theorem. Theorem 3.6. For α ∈ (0, 1] and λ = (λ1 , . . . , λn )0 , µ = (µ1 , . . . , µn )0 ∈ (0, ∞)n with λ w µ, let X i ∼ Γ (α, λi ) and Yi ∼ Γ (α, µi ), i ∈ {1, 2, . . . , n}. Then, (X 1:n , X 2:n , . . . , X n:n )0 ≥st (Y1:n , Y2:n , . . . , Yn:n )0 . This result was also proved by Lihong and Xinsheng [24] for the majorization order; see also [32, p. 319 and Example 6.B.26]. Now, by applying Theorem 3.3, we can easily extend Theorems 3.5 and 3.6 to the case of progressively Type-II censored order statistics. Corollary 3.7. For α ∈ (0, 1] and λ = (λ1 , . . . , λn )0 , µ = (µ1 , . . . , µn )0 ∈ (0, ∞)n with λ w µ, let either (1) X i ∼ Weibull(α, λi ) and Yi ∼ Weibull(α, µi ), i ∈ {1, 2, . . . , n}, or (2) X i ∼ Γ (α, λi ) and Yi ∼ Γ (α, µi ), i ∈ {1, 2, . . . , n}. Then, R R R R R R (X 1:m:n , X 2:m:n , . . . , X m:m:n )0 ≥st (Y1:m:n , Y2:m:n , . . . , Ym:m:n )0 .

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An extension of these results to the proportional hazards model is also possible, which covers the results of [30]; see also [12,29] for the univariate stochastic order. Definition 3.8. Let F = {F1 , . . . , Fn } be a set of distribution functions. F is said to have proportional hazard functions if for all x ∈ R, we have Fi (t) = 1 − e−λi H (t) ,

i = 1, 2, . . . , n,

where λ1 , . . . , λn > 0 and H : R −→ R is a non-decreasing function with H (t) = 0 for all t < 0 and limt→∞ H (t) = ∞. The parameters λ1 , . . . , λn are said to be constants of proportionality for the common hazard function H . We now present the corresponding ordering result by using the stronger version of [16,17]. Theorem 3.9. Suppose (1) F1 , F2 , . . . , Fn have proportional hazard functions with constants of proportionality λ1 , λ2 , . . . , λn > 0 for the common hazard function H , (2) G 1 , G 2 , . . . , G n have proportional hazard functions with constants of proportionality µ1 , µ2 , . . . , µn > 0 for the common hazard function H , and (3) (λ1 , λ2 , . . . , λn )0 w (µ1 , µ2 , . . . , µn )0 . Then, R R R R R R (X 1:m:n , X 2:m:n , . . . , X m:m:n )0 ≥st (Y1:m:n , Y2:m:n , . . . , Ym:m:n )0 .

The preceding result can be directly applied to extend a result of [22] on stochastic comparisons of spacings from INID exponential distributions [see also [21]]. Noting that the first progressively censored order statistics in a sample equals the minimum of the sample the proof is along the lines of Theorem 3.1 in [22] and therefore omitted. Corollary 3.10. Let X 1 , . . . , X n be independent exponential random variables with X i having hazard rate λi , i = 1, . . . , n. Let Y1 , . . . , Yn be another set of independent exponential random variables with hazard rate µ1 , . . . , µn , respectively. If (λ1 , λ2 , . . . , λn )0  (µ1 , µ2 , . . . , µn )0 then R R R R R R R R (X 2:m:n − X 1:m:n , . . . , X m:m:n − X 1:m:n )0 ≥st (Y2:m:n − Y1:m:n , . . . , Ym:m:n − Y1:m:n )0 . R R R R R Note that X 1:m:n and (X 2:m:n − X 1:m:n , . . . , X m:m:n − X 1:m:n )0 are independent in the exponential case [see [8] and, for order statistics, [21]].

Remark 3.11. The following observations are worth making from the above results: 1. An interesting special case of Corollary 3.7 and Theorem 3.9 is given by the particular choice P µi = n1 nj=1 λ j = λ¯ , i ∈ {1, 2, . . . , n}. Then, Y1 , . . . , Yn are IID random variables. In this R R case, the random vector (Y1:m:n , . . . , Ym:m:n )0 is stochastically smallest, since (λ1 , λ2 , . . . , λn )0  (λ¯ , λ¯ , . . . , λ¯ )0 applies to all (λ1 , . . . , λn )0 ∈ (0, ∞)n . Since the weak supermajorization order is used in ¯ ∞). For order statistics, such the above theorems, we can replace λ¯ by any value µ ∈ [λ, comparisons have been considered by Ball [10], Barbour et al. [11], Li and Shaked [23], and Ma [25].

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2. A similar situation is considered in Theorems 4.1 and 4.2 wherein the IID case is compared to the INID case. 3. Using a result of [13], Pmunder the conditions of the preceding results for any censoring scheme (R1 , . . . , Rm ) with i=1 Ri = n − m, we obtain R R R (X 1:m:n , X 2:m:n , . . . , X m:m:n )0 ≥st (Y1:n , . . . , Ym:n )0 ,

where Y1:n , . . . , Ym:n are the first m order statistics based on an IID sample. 4. A very interesting result has been established by Ma [25] when the X -sample is INID and the Y -sample is IID. He proved that the stochastic order of the order statistics is determined by the stochastic ordering of the minimum and maximum of the samples. Thus, Theorem 3.3 can be extended as follows: R R R R X 1:n ≥st Y1:n yields (X 1:m:n , . . . , X m:m:n )0 ≥st (Y1:m:n , . . . , Ym:m:n )0

and R R R R )0 ≤st (Y1:m:n , . . . , Ym:m:n )0 . X n:n ≤st Yn:n yields (X 1:m:n , . . . , X m:m:n

The following result is a direct consequence of Corollary 3.7 and Theorems 3.4 and 3.9. Corollary 3.12. Under the conditions of either Corollary 3.7, Theorem 3.4 or Theorem 3.9, for any c1 , . . . , cm ≥ 0, we have m X

R ci X i:m:n ≥st

i=1

m X

R ci Yi:m:n .

i=1

In particular, this yields k X

R X i:m:n ≥st

i=1

k X

R Yi:m:n ,

k ∈ {1, 2, . . . , m}.

i=1

4. Extension of Sen’s inequalities As a final application of the mixture representation in Theorem 2.2, we extend a result of [31] [see also [7]] to the case of progressively Type-II censored order statistics. In contrast to the preceding results, no restrictions are imposed here on the underlying distribution functions F1 , . . . , Fn . Subsequently, we use the notations from Section 3. For the usual order statistics in the INID case, Sen [31] established the following result. Theorem 4.1 ([31]). Let Gi ≡ G =

n 1X ¯ Fk = F, n k=1

i ∈ {1, 2, . . . , n}

¯ ¯ p ∈ (0, 1). Then, for and ξ p = inf{u ∈ R : F(u) ≥ p} be the p-quantile of F, i ∈ {2, 3, . . . , n − 1} and x ≤ ξ i−1 < ξ i ≤ y, we have n

n

P (x < X i:n ≤ y) ≥ P (x < Yi:n ≤ y)

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¯ ¯ with equality holding iff Fk (x) = F(x) and Fk (y) = F(y) for all k ∈ {1, 2, . . . , n}. Furthermore, for x ∈ R, we have P (X 1:n ≤ x) ≥ P (Y1:n ≤ x)

and

P (X n:n ≤ x) ≤ P(Yn:n ≤ x),

¯ with equality holding iff Fk (x) = F(x) for all k ∈ {1, 2, . . . , n}. The analogue of Theorem 4.1 for progressively Type-II censored order statistics in the INID case is as follows. Theorem 4.2. Let Gi =

n 1X ¯ Fk = F, n k=1

αi = i − 1,

i ∈ {1, 2, . . . , n},

βi = n − γi + 1,

i ∈ {2, 3, . . . , m}

¯ ¯ p ∈ (0, 1). Then, for and ξ p = inf{u ∈ R : F(u) ≥ p} be the p-quantile of F, i ∈ {2, 3, . . . , m − 1} and x ≤ ξ αi < ξ βi ≤ y, we have n





n



 R R P x < X i:m:n ≤ y ≥ P x < Yi:m:n ≤y , ¯ ¯ where equality holds if Fk (x) = F(x) and Fk (y) = F(y) for all k ∈ {1, 2, . . . , n}. If Rm > 0, we additionally have     R R P x < X m:m:n ≤ y ≥ P x < Ym:m:n ≤y ∀x ≤ ξ αm < ξ βm ≤ y n

n

¯ ¯ with equality holding if Fk (x) = F(x) and Fk (y) = F(y) for all k ∈ {1, 2, . . . , n}. Furthermore, for x ∈ R, we have     R R P X 1:m:n ≤ x ≥ P Y1:m:n ≤x ¯ with equality holding iff Fk (x) = F(x) for all k ∈ {1, 2, . . . , n}. Proof. From the mixture representation in Theorem 2.2 (see Remark 2.6), we have for i ∈ {2, 3, . . . , m} R P(X i:m:n ≤ x) =

1 m−1 Q  l=1

X

γl −1 Rl



P(X ki :n ≤ x),

x ∈ R,

ZR

where all ki occurring in the summation are elements of the set (see Construction 2.1) {i, i + 1, . . . , n − γi + 1} . Note that i < m implies ki < n; if i = m, the inequality km < n follows from Rm > 0. Thus, the maximum is not included in the mixture representation so that Theorem 4.1 can be applied. Finally, for all i ∈ {2, 3, . . . , m} and l ∈ {i, i + 1, . . . , n − γi + 1}, we have ξ αi ≤ ξ l−1 n

n

as well as ξ βi ≥ ξ l n

n

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from which the result follows immediately. Let i ∈ {2, 3, . . . , m − 1} and x ≤ ξ αi < ξ βi ≤ y; n

then, R P(x < X i:m:n ≤ y) =

1 m−1 Q  l=1

≥

γl −1 Rl

1  m−1 Q l=1

γl −1 Rl

X 

n

P(x < X ki :n ≤ y)

ZR

X 

P(x < Yki :n ≤ y)

ZR

R = P(x < Yi:m:n ≤ y).

¯ ¯ The equality conditions Fk (x) = F(x) and Fk (y) = F(y), k ∈ {1, 2, . . . , n}, also follow from Theorem 4.1. If i = m and Rm > 0, the result follows analogously. Since R X 1:m:n ≡ X 1:n

and

R Y1:m:n ≡ Y1:n

the proof of the theorem is complete.

C

Remark 4.3. The following points are worth noting here: 1. In Theorem 4.1, the equality conditions are if and only if conditions. Due to use of the mixture representation in Theorem 2.2, we obtain here only if conditions for the case of progressively Type-II censored order statistics. 2. The condition that Rm > 0 cannot be dropped without replacing it by another assumption because, in this case, km may be equal to n. Since in the case of the maximum X n:n the inequality is reversed, the result cannot be established in that way (see Theorem 4.1). References [1] S. Alvarez-Andrade, N. Balakrishnan, L. Bordes, Homogeneity tests based on several progressively Type-II censored samples, J. Multivariate Anal. 98 (2007) 1195–1213. [2] B.C. Arnold, N. Balakrishnan, Relations, Bounds and Approximations for Order Statistics, in: Lecture Notes in Statistics, vol. 53, Springer, Berlin, 1989. [3] I. Bairamov, Progressive type II censored order statistics for multivariate observations, J. Multivariate Anal. 97 (2006) 797–809. [4] N. Balakrishnan, Permanents, order statistics, outliers, and robustness, Rev. Mat. Complut. 20 (2007) 7–107. [5] N. Balakrishnan, Progressive censoring methodology: an appraisal (with Discussions), Test 16 (2007) 211–296. [6] N. Balakrishnan, R. Aggarwala, Progressive Censoring: Theory, Methods, and Applications, Birkh¨auser, Boston, 2000. [7] N. Balakrishnan, K. Balasubramanian, Revisiting Sen’s inequalities on order statistics, Statist. Probab. Letters (2007), in press (doi:10.1016/j.spl.2007.09.023). [8] N. Balakrishnan, E. Cramer, Progressive censoring from heterogeneous distributions with applications to robustness, Ann. Inst. Statist. Math. 60 (2007) 151–171. [9] N. Balakrishnan, A. Dembinska, Progressively Type-II right censored order statistics from discrete distributions, J. Statist. Plann. Inference 138 (2007) 845–856. [10] F. Ball, Deterministic and stochastic epidemics with several kinds of susceptibles, Adv. Appl. Probab. 17 (1985) 1–22. [11] A. Barbour, T. Lindvall, L. Rogers, Stochastic ordering of order statistics, J. Appl. Probab. 28 (1991) 278–286. [12] P.J. Boland, M. Shaked, J.G. Shanthikumar, Stochastic ordering of order statistics, in: N. Balakrishnan, C.R. Rao (Eds.), Order Statistics: Theory and Methods, Handbook of Statistics, vol. 16, Elsevier, Amsterdam, 1998, pp. 89–103. [13] M. Burkschat, Estimation with Generalized Order Statistics, Ph.D. Thesis, RWTH Aachen University, 2006.

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