Transform orders and stochastic monotonicityof statistical functionals

Abstract

In some inferential statistical methods, such as testsand confidence intervals, it is important to describe thestochastic behavior of statistical functionals, aside fromtheir large sample properties. We study such a behaviorin terms of the usual stochastic order. For this purpose,we introduce a generalized family of stochastic orders,which is referred to as transform orders, showing thatit provides a flexible framework for deriving stochasticmonotonicity results. Given that our general definitionmakes it possible to obtain some well known order-ing relations as particular cases, we can easily applyour method to different families of functionals. Theseinclude some prominent inequality measures, such asthe generalized entropy, the Gini index, and its gen-eralizations. We also illustrate the applicability of ourapproach by determining the least favorable distribu-tion, and the behavior of some bootstrap statistics, insome goodness-of-fit testing procedures.