Integrated quantile functions: properties and applications

Alexander A Gushchin,Dmitriy A Borzykh

doi:10.15559/17-vmsta88

Abstract

In this paper we provide a systematic exposition of basic properties of integrated distribution and quantile functions. We define these transforms in such a way that they characterize any probability distribution on the real line and are Fenchel conjugates of each other. We show that uniform integrability, weak convergence and tightness admit a convenient characterization in terms of integrated quantile functions. As an application we demonstrate how some basic results of the theory of comparison of binary statistical experiments can be deduced using integrated quantile functions. Finally, we extend the area of application of the Chacon--Walsh construction in the Skorokhod embedding problem.

Highlights

Integrated distribution and quantile functions or simple transformations of them play an important role in probability theory, mathematical statistics, and their applications www.i-journals.org/vmstaA.A
In this paper we provide a systematic exposition of basic properties of integrated distribution and quantile functions
We show that uniform integrability, weak convergence and tightness admit a convenient characterization in terms of integrated quantile functions

Summary

Introduction

Integrated distribution and quantile functions or simple transformations of them play an important role in probability theory, mathematical statistics, and their applications. One of the key points of our approach is that we define integrated distribution and quantile functions as Fenchel conjugates of each other. The integrated quantile function of a random variable X, as it is defined in our paper, is a convex function whose gradient pushes forward the uniform distribution on (0, 1) into the distribution of X; the integrated distribution function is the Fenchel transform of the integrated quantile function and its gradient pushes forward the distribution of X into the uniform distribution on (0, 1) if the distribution of X is continuous It the multidimensional case the existence of such functions follows from the McCann theorem [18].

Definition and properties of integrated distribution functions

Convex orders

Examples

Uniform integrability and weak convergence

Applications to binary statistical models

Chacon–Walsh revisited