Speeches

14 views

Dependence orderings for generalized order statistics

Dependence orderings for generalized order statistics
of 11

Please download to get full document.

View again

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Share
Transcript
  Statistics & Probability Letters 73 (2005) 357–367 Dependence orderings for generalized order statistics Baha-Eldin Khaledi a,  , Subhash Kochar b a Department of Statistics, College of Sciences, Kermanshah, Iran b Indian Statistical Institute, 7, SJS Sansanwal Marg, New Delhi 110016, India Received 23 June 2004; received in revised form 10 March 2005Available online 13 May 2005 Abstract Generalized order statistics (gOSs) unify the study of order statistics, record values,  k  -records, Pfeifer’srecords and several other cases of ordered random variables. In this paper we consider the problem of comparing the degree of dependence between a pair of gOSs thus extending the recent work of Ave ´rous etal. [2005. J. Multivariate Anal. 94, 159–171]. It is noticed that as in the case of ordinary order statistics,copula of gOSs is independent of the parent distribution. For this comparison we consider the notion of  more regression dependence or more stochastic increasing . It follows that under some conditions, for  i  o  j  , thedependence of the  j  th generalized order statistic on the  i  th generalized order statistic decreases as  i   and  j  draw apart. We also obtain a closed-form expression for Kendall’s coefficient of concordance between apair of record values. r 2005 Elsevier B.V. All rights reserved. Keywords:  Dispersive ordering; Pure birth process; Exponential distribution; Kendall’s tau; Monotone regressiondependence; Stochastic increasingness; Record values 1. Introduction Order statistics and record values play an important role in statistics, in general, and inReliability Theory and Life Testing, in particular. Their distributional and stochastic propertieshave been studied extensively but separately in the literature. However, they can be considered as ARTICLE IN PRESS www.elsevier.com/locate/stapro 0167-7152/$-see front matter r 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.spl.2005.04.023  Corresponding author. Work done while visiting Indian Statistical Institute, Delhi Center E-mail addresses:  bkhaledi@hotmail.com (B.-E. Khaledi), kochar@isid.ac.in (S. Kochar).  special cases of generalized order statistics (gOSs) (cf. Kamps, 1995) which in addition coverparticular sequential order statistics,  k  th record values, Pfeifer’s record model,  k  n  record fromnonidentical distributions, and ordered random variables which arise from truncated distribu-tions. It is well known that a sequence of record values can be viewed as a sequence of theoccurrence times of a certain nonhomogeneous Poisson process. It is also connected to the failuretimes of a minimal repair process. There is a close connection between Pfeifer’s records and theoccurrence times of a pure birth process (cf. Pfeifer, 1982a,b).As mentioned above, many interesting stochastic ordering results for order statistics andspacings on the one hand, and for record values and record increments on the other hand, havebeen obtained separately by many investigators without realizing that perhaps they can be unifiedunder the umbrella of gOSs. Kamps (1995) in the last chapter of his book studied some reliabilityproperties of gOSs. Franco et al. (2002) obtained some stochastic ordering results for spacingsof gOSs.Recently Ave ´rous et al. (2005) have studied the dependence properties of order statistics of arandom sample from a continuous distribution. To compare the degree of association betweentwo such pairs of ordered random variables, they considered a notion of relative monotoneregression dependence (or stochastic increasingness). Using this concept, they proved that for  i  o  j  ,the dependence of the  j  th order statistic on the  i  th order statistic decreases as  i   and  j   draw apart. Inthis paper we study dependence properties of a pair of gOSs and as a consequence these resultswill be applicable to order statistics, record values, occurrence times of a pure birth process, andall those models which are covered under gOSs.The organization of the paper is as follows. In Section 2, we introduce gOSs and state the maintheorem which describes the conditions under which a pair of gOSs is more dependent thananother pair in the sense of   more SI   ordering. It is seen that the work of  Ave ´rous et al. (2005) canbe extended to the gOSs. In Section 3 we point out a close connection that exists between theconcepts of dispersive ordering and that of   more SI   ordering. The proofs of the various resultsare given in this section. In the last section, we obtain a closed-form expression for the value of theKendall’s  t  between a pair of record values. 2. Main results First we give the definition of the joint distribution of   n  gOSs (cf. Kamps, 1995, p. 49). Definition 2.1.  Let  n  2  N  ,  k  X 1,  m 1 ;  . . .  ; m n  1  2 R ,  M  r  ¼ P n  1  j  ¼ r  m  j  , 1 p r p n  1 be parameterssuch that g r  ¼  k  þ n  r þ M  r X 1 for all  r  2 f 1 ;  . . .  ; n  1 g ,and let  ~ m  ¼ ð m 1 ;  . . .  ; m n  1 Þ , if   n X 2 (  ~ m  2 R  arbitrary, if   n  ¼  1).If the random variables  U  ð r ; n ;  ~ m ; k  Þ ,  r  ¼  1 ;  . . .  ; n , possess a joint density function of the form  f  U  ð 1 ; n ;  ~ m ; k  Þ ; ... ; U  ð n ; n ;  ~ m ; k  Þ ð u 1 ;  . . .  ; u n Þ ¼  k  Y n  1  j  ¼ 1 g  j   ! Y n  1  j  ¼ 1 ð 1  u i  Þ m i   ! ð 1  u n Þ k   1 on the cone 0 p u 1 p  p u n o 1 of   R n , then they are called  uniform gOSs . ARTICLE IN PRESS B.-E. Khaledi, S. Kochar / Statistics & Probability Letters 73 (2005) 357–367  358  Generalized order statistics based on an arbitrary continous distribution with distributionfunction  F   are now defined by means of the quantile transformation X  ð r ; n ;  ~ m ; k  Þ ¼  F   1 ð U  ð r ; n ;  ~ m ; k  ÞÞ ;  r  ¼  1 ;  . . .  ; n ,and they are denoted by gOSs. As discussed by Kamps (1995), for suitable choices of theparameters these reduce to the joint distributions of order statistics from a continuousdistribution, record values, Pfeifer’s record values, and so on.Let  ð S  ; T  Þ  be a continuous bivariate random vector with joint distribution function  H  . Recallthat  T   is said to be stochastically increasing in  S   if and only if, for all  s ; s 0 ; t  2 R , s p s 0 ¼)  P  ð T  p t j S   ¼  s 0 Þ p PT  p t j S   ¼  s Þ . (2.1)Let  H  ½ s   denote the distribution function of the conditional distribution of   T   given  S   ¼  s . Theabove implication may then be expressed in the alternate form s p s 0 ¼)  H  ½ s 0    H   1 ½ s   ð u Þ p u ,where  u  2 ð 0 ; 1 Þ . Note that property (2.1) is not symmetric in  S   and  T  , but that in case thesevariables are independent,  H  ½ s 0    H   1 ½ s   ð u Þ   u  for all  u  2 ð 0 ; 1 Þ  and for all  s ; s 0 2 R . Observe alsothat if   x  p  ¼  F   1 ð  p Þ  denotes the  p th quantile of the marginal distribution of   S  , then (2.1) isequivalent to the condition0 o  p p q o 1  ¼)  H  ½ x q    H   1 ½ x  p  ð u Þ p u holding true for all  u  2 ð 0 ; 1 Þ .To compare the relative degree of dependence between arbitrary pairs of gOSs we use thenotion of   more stochastically increasing  dependence ordering as discussed by Ave ´rous et al. (2005).For  i   ¼  1 ; 2, let  ð S  i  ; T  i  Þ  be a pair of continuous random variables with joint cumulativedistribution function  H  i   and marginals  F  i   and  G  i  . Definition 2.2.  T  2  is said to be more stochastically increasing in  S  2  than  T  1  is in  S  1 , denoted by ð T  1 j S  1 Þ SI ð T  2 j S  2 Þ  or  H  1  SI H  2 , if and only if 0 o  p p q o 1  ¼)  H  2 ½ x 2 q    H   12 ½ x 2  p  ð u Þ p H  1 ½ x 1 q    H   11 ½ x 1  p  ð u Þ , (2.2)for all  u  2 ð 0 ; 1 Þ , where for  i   ¼  1 ; 2,  H  i  ½ s   denotes the conditional distribution of   T  i   given  S  i   ¼  s ,and  x ip  ¼  F   1 i   ð  p Þ  stands for the  p th quantile of the marginal distribution of   S  i  .Obviously, (2.2) implies that  T  2  is stochastically increasing in  S  2  if   S  1  and  T  1  are independent.It also implies that if   T  1  is stochastically increasing in  S  1 , then so is  T  2  in  S  2 ; and conversely, if   T  2 is stochastically decreasing in  S  2 , then so is  T  1  in  S  1 . As observed by Ave ´rous et al. (2005), theabove definition of more SI ordering depends on the joint distributions of the underlying randomvariables only through their copulas. Also, ð T  1 j S  1 Þ SI ð T  2 j S  2 Þ )  C  1 ð u ; v Þ p C  2 ð u ; v Þ , (2.3)where  C  i   is the copula of   ð S  i  ; T  i  Þ ,  i   ¼  1 ; 2, which in turn implies that k ð S  1 ; T  1 Þ p k ð S  2 ; T  2 Þ ,where  k ð S  ; T  Þ represents Spearman’s  r , Kendall’s  t , Gini’s coefficient, or indeed any other copula-based measure of concordance satisfying the axioms of  Scarsini (1984). In the special case where ARTICLE IN PRESS B.-E. Khaledi, S. Kochar / Statistics & Probability Letters 73 (2005) 357–367   359  F  1  ¼  F  2  and  G  1  ¼  G  2 , it also follows from (2.3) that the pairs  ð S  1 ; T  1 Þ  and  ð S  2 ; T  2 Þ  are ordered byPearson’s correlation coefficient, namely,corr ð S  1 ; T  1 Þ p corr ð S  2 ; T  2 Þ .Note that the copula of a pair of gOSs is independent of the parent distribution  F  . For comparingtwo different gOSs we use the following pre-ordering on  R þ n . Definition 2.3.  A vector  x  in  R þ n is said to be  p -larger than another vector  y  also in  R þ n (written x   p y ) if  Q  j i  ¼ 1  x ð i  Þ p Q  j i  ¼ 1  y ð i  Þ ;  j   ¼  1 ;  . . .  ; n , where  x ð 1 Þ p  p x ð n Þ  and  y ð 1 Þ p  p  y ð n Þ  are theincreasing arrangements of the components of   x  and  y , respectively.Now we state the main theorem of this paper whose proof is given in Section 3. Theorem 2.1.  Let  ð X  ð r ; n ;  ~ m ; k  Þ ;  r  ¼  1 ;  . . .  ; n Þ  and   ð X  0 ð r 0 ; n 0 ;  ~ m 0 ; k  0 Þ  r  ¼  1 ;  . . .  ; n Þ  be the  gOSs  based on distributions F and G  ,  respectively. Let  g r  ¼  k  þ n  r þ P n  1 h ¼ r m h  and   g 0 r  ¼ k  0 þ n 0  r þ P n  1 h ¼ r m 0 h . Then for i  p  j and i  0 p  j  0 , ð X  0 ð  j  0 ; n 0 ;  ~ m 0 ; k  0 Þ j  X  0 ð i  0 ; n 0 ;  ~ m 0 ; k  0 ÞÞ SI ð X  ð  j  ; n ;  ~ m ; k  Þ j  X  ð i  ; n ;  ~ m ; k  ÞÞ ,  provided the following conditions are satisfied  :(a1)  i  X i  0 and j    i  p  j  0  i  0 .(a2)  ð g ‘  1 ;  . . .  ; g ‘  i  0 Þ  p ð g 0 1 ;  . . .  ; g 0 i  0 Þ  for some set  f ‘  1 ;  . . .  ; ‘  i  0 g  f 1 ;  . . .  ; i  g .(a3)  ð g 0 k  1 ;  . . .  ; g 0 k   j   i  Þ  p ð g i  þ 1 ;  . . .  ; g  j  Þ  for some set  f k  1 ;  . . .  ; k   j   i  g  f i  0 þ 1 ;  . . .  ;  j  0 g .It is well known that for specific sets of parameters,  n ,  k  , and  m i  ,  i   ¼  1 ;  . . .  ; n  1, thegOSs reduce to the well-known ordered random variables. Now we find sufficient conditions onthe parameters of the various sub-models of gOSs for which Theorem 2.1 holds.(A)  Order statistics from i.i.d random variables : For  n X 1, let  X  i  : n  denote the  i  th order statisticbased on a random sample  X  1 ;  . . .  ; X  n  from a continuous distribution with cdf   F  . This is a specialcase of gOSs with  m 1  ¼  ¼  m n  1  ¼  0 and  k   ¼  1. In this case  g r  ¼  n  r þ 1,  r  ¼  1 ;  . . .  ; n  1.Let  m i   ¼  m 0 i   ¼  0,  i   ¼  1 ;  . . .  ; n  1 and  k   ¼  k  0 ¼  1. With these settings we see that the conditions(a2) and (a3) are satisfied when  n  i  p n 0  i  0 and  n   j  X n 0   j  0 . That is, for  i  X i  0 ,  j    i  p  j  0  i  0 , n  i  p n 0  i  0 , and  n   j  X n 0   j  0 , we have ð X  0  j  0 : n 0  j  X  0 i  0 : n 0 Þ SI ð X   j  : n  j  X  i  : n Þ ,as proved recently by Ave ´rous et al. (2005). In the special case of one-sample problem when n  ¼  n 0 , we have the following results:(a)  ð X  k  : n j  X  i  : n Þ SI ð X   j  : n j X  i  : n Þ  for all 1 p i  o  j  o k  p n ,(b)  ð X   j  : n j  X  i  : n Þ SI ð X   j  þ 1 : n þ 1 j X  i  þ 1 : n þ 1 Þ  for all 1 p i  o  j  p n ,(c)  ð X  n þ 1 : n þ 1 j X  1 : n þ 1 Þ SI ð X  n : n j  X  1 : n Þ  for every integer  n X 2.(B)  k-Records : Let  f X  i  ;  i  X 1 g  be a sequence of i.i.d random variables from a continuousdistribution  F   and let  k   be a positive integer. The random variables  L ð k  Þ ð n Þ  given by  L ð k  Þ ð 1 Þ ¼  1, L ð k  Þ ð n þ 1 Þ ¼  min f  j   2  N  ; X   j  :  j  þ k   1 4 X  L ð k  Þ ð n Þ : L ð k  Þ ðð n Þþ k   1 Þ g ;  n X 1, ARTICLE IN PRESS B.-E. Khaledi, S. Kochar / Statistics & Probability Letters 73 (2005) 357–367  360  are called the  n th  k  -record times and the quantities  X  L ð k  Þ ð n Þ : L ð k  Þ ðð n Þþ k   1 Þ  which we denote by  R ð n  :  k  Þ are termed the  n th  k  -records (cf. Kamps, 1995, p. 34 and Arnold et al., 1998). The joint density of  the first  n k  -records corresponding to a sequence of independent random variables from acontinuous distribution  F   is a special case of the joint density of first  n  gOSs with m 1  ¼  ¼  m n  1  ¼  1. In this case  g r  ¼  k  ,  r  ¼  1 ;  . . .  ; n  1. Now let  m i   ¼  m 0 i   ¼  1, i   ¼  1 ;  . . .  ; n  1, and  k   ¼  k  0 . Using the above setting it follows that conditions (a2) and (a3) of Theorem 2.1 are satisfied. Therefore, for  i  X i  0 ,  j    i  p  j  0  i  0 , we have ð R 0 ð  j  0 :  k  Þ j  R 0 ð i  0 :  k  ÞÞ SI ð R ð  j   :  k  Þ j  R ð i   :  k  ÞÞ ,where  R ð  j   :  k  Þ ,  j  X 1 and  R 0 ð  j  0 :  k  Þ ,  j  0 X 1 stand for the  j  th and  j  0 th  k  -records. This means that for i  o  j  , the dependence of the  j  th  k  -record on the  i  th  k  -record decreases as  i   and  j   draw apart.(C)  Two-stage progressive type II censoring : Let  X  1 ;  . . .  ; X  v  be a random sample from acontinuous distribution  F  . Let these be the lifetimes of   v  items put on test at time  t  ¼  0. At thetime of the  r 1 th failure,  n 1  functioning items are randomly selected and removed from the test. Thetest terminates when further  r 2  items have failed. The  n  ¼  r 1  þ r 2  observations  X  1 : v p  p X  n : v are called order statistics arising in progressive type II censoring with two stages. This is a specialcase of gOSs with  m 1  ¼  ¼  m r 1  1  ¼  m r 1 þ 1  ¼  ¼  m n  1  ¼  0,  m r 1  ¼  n 1  and  k   ¼  v  n 1   n þ 1.In this case  g r  ¼  v  r þ 1,  r  ¼  1 ;  . . .  ; r 1  and  g r  ¼  v  n 1   r þ 1,  r  ¼  r 1  þ 1 ;  . . .  ; n  1. Let m i   ¼  m 0 i   ¼  0,  i   ¼  1 ;  . . .  ; r 1   1 ; r 1  þ 1 ;  . . .  ; n  1,  m r 1  ¼  m 0 r 1 ¼  n 1 ,  k   ¼  v  n 1   n þ 1 and k  0 ¼  v 0  n 1   n þ 1. With these settings we see that conditions (a2) and (a3) are satisfied when v  i  p v 0  i  0 and  v   j  X v 0   j  0 . That is, for  i  X i  0 ;  j    i  p  j  0  i  0 , v  i  p v 0  i  0 and  v   j  X v 0   j  0 ,we have ð X  0  j  0 : v 0  j  X  0 i  0 : v 0 Þ SI ð X   j  : v  j  X  i  : v Þ .As discussed by Kamps (1995), there are many other models like Pfeifer’s records, sequentialorder statistics, order statistics with nonintegral sample size, etc. which can also be expressed asspecial cases of gOSs. 3. Auxiliary results and proofs In this section we prove some auxiliary results to prove our Theorem 2.1. As we will see, there isa close connection between the concepts of dispersive ordering and  more SI ordering . Definition 3.1.  A random variable  X   with distribution function  F   is said to be less dispersed thananother variable  Y   with distribution  G  , written as  X  p disp Y   or  F  p disp G  , if and only if  F   1 ð b Þ F   1 ð a Þ p G   1 ð b Þ G   1 ð a Þ for all 0 o a p b o 1.It is easy to see that the  F  p disp G   is equivalent to F  f F   1 ð u Þ c g p G  f G   1 ð u Þ c g  for every  c X 0 and  u  2 ð 0 ; 1 Þ .For general information about dispersive ordering and its properties, refer to Shaked andShanthikumar (1994, Section 2.B). The next proposition establishes a close connection betweendispersive ordering and  more SI ordering . ARTICLE IN PRESS B.-E. Khaledi, S. Kochar / Statistics & Probability Letters 73 (2005) 357–367   361
Advertisement
Related Documents
View more
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks