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Stochastic orders and dependence properties of concomitants of order statistics
Given a bivariate sample Xi,Yi ni=1 , the rth order statistic Xr is the rth smallest value of the X's and the rth concomitant Yr is the Y value that accompanies Xr . 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 Y&sqbl0;&sqbr0; is multivariate associated. If the conditional hazard rate function of Y given X, fY&vbm0;Xy &bsolm0;x /F&d1;Y&vbm0;X&parl0;y &bsolm0; x&parr0;, is decreasing in x, Y&sqbl0;&sqbr0; is multivariate right corner set increasing. If X and Y are totally positive dependent of order 2, then Y&sqbl0;&sqbr0; is multivariate totally positive dependent of order 2, and the univariate concomitants Yr 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 Xr and Xs 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.^
Todd David Blessinger,
"Stochastic orders and dependence properties of concomitants of order statistics"
(January 1, 1999).
Electronic Doctoral Dissertations for UMass Amherst.