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Stochastic orders and dependence properties of concomitants of order statistics
Abstract
Given a bivariate sample [special characters omitted], the rth order statistic [special characters omitted] is the rth smallest value of the X's and the rth concomitant [special characters omitted] is the Y value that accompanies [special characters omitted]. Order statistics are widely used, and their stochastic order and dependence properties have been studied extensively. In this dissertation, we show that if X and Y are positively dependent, then the concomitants satisfy certain stochastic order relations and positive dependence properties as well, at least in the case where the vectors ( Xi, Yi) are independent and identically distributed and come from an absolutely continuous distribution. It, is shown that if Y is stochastically increasing in X, the concomitants increase in multivariate stochastic order, and the entire vector of concomitants [special characters omitted] is multivariate associated. If the conditional hazard rate function of Y given X, [special characters omitted] is decreasing in x, [special characters omitted] is multivariate right corner set increasing. If X and Y are totally positive dependent of order 2, then [special characters omitted] is multivariate totally positive dependent of order 2, and the univariate concomitants [special characters omitted] increase in likelihood ratio order as r increases. Concomitants have not previously been studied in the discrete case much because, unlike the continuous case, the probability that two order statistics [special characters omitted] and [special characters omitted] are equal is positive; thus, the concomitants are not immediately determinable. Here we introduce a way of assigning concomitants in the discrete case and prove two related results for that case.
Subject Area
Statistics|Mathematics
Recommended Citation
Blessinger, Todd David, "Stochastic orders and dependence properties of concomitants of order statistics" (1999). Doctoral Dissertations Available from Proquest. AAI9932295.
https://scholarworks.umass.edu/dissertations/AAI9932295