Transform orders and stochastic monotonicity of statistical functionals

TL;DR

This paper introduces transform orders, a new generalized stochastic order framework, to analyze the stochastic behavior of statistical functionals, including inequality measures and bootstrap statistics, in inferential methods.

Contribution

It proposes a flexible, unified approach to derive stochastic monotonicity results for various statistical functionals using transform orders.

Findings

Derived stochastic monotonicity for inequality measures like Gini index

Applied the framework to bootstrap statistics in goodness-of-fit tests

Identified least favorable distributions using the new approach

Abstract

In some inferential statistical methods, such as tests and confidence intervals, it is important to describe the stochastic behavior of statistical functionals, aside from their large sample properties. We study such behavior in 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 that it provides a flexible framework for deriving stochastic monotonicity results. Given that our general definition makes it possible to obtain some well-known ordering relations as particular cases, we can easily apply our method to different families of functionals. These include some prominent inequality measures, such as the generalized entropy, the Gini index, and its generalizations. We also illustrate the applicability of our approach by determining the least favorable distribution, and the behavior of some bootstrap statistics, in some goodness-of-fit testing procedures.