Multivariate order statistics based on dependent and nonidentically distributed random variables

1. Joint binomial moments (bivariate case) The last two decades have seen major developments in the theory of order statistics and its applications to practical problems. Under the impetus of advances in the probabilistic theory, new statistical methods have been developed for both univariate and multivariate problems. The books by [6,1,5] are considered among the most popular books on this topic of research. They pay special attention to univariate order statistics, their properties, inferential issues, and more importantly their applications to practical problems. The book by [6] provides detailed description of order statistics of multivariate observations. For a recent paper on this topic, see [2]. The early applications of order statistics to practical problems were based on the assumption that the basic random variables (rv’s) are i.i.d. The need for deviating from this rarely correct assumption has been stressed ever since the theory was applied to specific problems. For example, in the study of the air pollution, if we let Xj be the concentration of a pollutant in the jth time interval of a predetermined length, it is reasonable to assume that the Xj are identically distributed but successive Xj values are dependent. Also, the approximation by i.i.d. variables is not justified in describing the random time to first failure of a piece of equipment in the general case. However, in any univariate or multivariate practical problem even if the approximation by i.i.d. variables is good, the credibility of a solution should be questioned if it starts by assuming that the variables are i.i.d. when they are not. In this section we start with deriving a general identity and inequalities which involve the joint distribution of order statistics in a set of dependent and nonidentical rv’s. Let A(1) = (A 1 , A (1) 2 , . . . , A (1) n1 ) and A(2) = (A 1 , A (2) 2 , . . . , A (2) n2 ) be two sequences of events on the same probability space. Let ν1 = νn1(A (1)) and ν2 = νn2(A (2)), respectively, be the number of those A j and A (2) j which occur. Let S0:n1;0:n2 = 1, Sk1:n1;0:n2 = S (1) k1:n1 and S0:n1;k2:n2 = S (2) k2:n2 , where E-mail address: [email protected]. 0047-259X/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jmva.2008.03.002 82 H.M. Barakat / Journal of Multivariate Analysis 100 (2009) 81–90 S ki:ni = ∑ 1≤j1 n1 or k2 > n2, when the sum above is empty and thus Sk1:n1;k2:n2 = 0 for at least one of the inequalities k1 > n1 and k2 > n2 holds. The problem of this section is to derive an exact formula for P(t1, t2) = Pr(ν1 = t1, ν2 = t2), as well as to set bounds on P(t1, t2) in terms of Sk1:n1;k2:n2 , 0 ≤ ki ≤ ni, i = 1, 2. The quantity Sk1:n1;k2:n2 is called the joint (k1, k2)th binomial moment of the vector (νn1(A (1)), νn2(A (2))) (cf. [6,7]). The study of P(t1, t2) will enable us to obtain exact expressions for, or bounds on, the distribution of the joint order statistics, which are based on a general sequence of rv’s (e.g., not necessarily independent or identical). We first present three lemmas; the first of them is due to [7]. Lemma 1.1. For any k1, k2 ≥ 0, we get Sk1:n1;k2:n2 = E (( ν1 k1 )( ν2 k2 )) = n1 ∑