Order Statistics of Dependent Observations

Assume that Y₁,…, Y _n are possibly dependent and identically distributed. Recalling arguments of Rychlik [79], in Section 5.1 we conclude sharp bounds (2.25) and (2.27) on expectations of general L-statistics and single order statistics, respectively, depending on the common marginal distribution of the observations. Next we apply the projections of functionals defined in (2.25) and (2.27) for establishing respective moment bounds over general and restricted families of marginals. In Section 5.2 general, symmetric, and nonnegative observations are treated, and respective deterministic bounds for arbitrary samples are concluded. The results cited are from Rychlik [81], but some earlier partial solutions are also mentioned. In the remainder of Chapter 5 we confine ourselves to single order statistics. In Section 5.3 we present mean-variance and second moment bounds on the expectations of order statistics for families of parent distributions related to a given one in the convex order. The mean-variance bounds for F$${\underline \succ _c}$$W and F$${\underline \prec _c}$$W are not published elsewhere, except for DFR and IFR distributions, given in Rychlik [87]. The results for general W and W = U (i.e., for the decreasing density distributions) are presented here as well. Rychlik [84] established second moment bounds for F$${\underline \succ _c}$$ ($${\underline \prec _c}$$)W with general W. The decreasing density and failure rate distributions were studied in Gajek and Rychlik [32].