Kullback-Leibler Information of Consecutive Order Statistics

Abstract

A calculation of the Kullback-Leibler information of consecutive order statistics is complicated because it depends on a multi-dimensional integral. Park (2014) discussed a representation of the Kullback-Leibler information of the first r order statistics in terms of the hazard function and simplified the r-fold integral to a single integral. In this paper, we first express the Kullback-Leibler information in terms of the reversed hazard function. Then we establish a generalized result of Park (2014) to an arbitrary consecutive order statistics. We derive a single integral form of the Kullback-Leibler information of an arbitrary block of order statistics; in addition, its relation to the Fisher information of order statistics is discussed with numerical examples provided.