A new convexity-based inequality, characterization of probability distributions and some free-of-distribution tests

TL;DR

This paper introduces a new convexity-based inequality involving probability distribution functions, which characterizes certain distributions and enables the construction of distribution-free two-sample tests.

Contribution

It presents a novel inequality based on convex functions, providing a new way to characterize distributions and develop free-of-distribution statistical tests.

Findings

New convexity-based inequality established

Characterization of specific probability distributions achieved

Development of distribution-free two-sample tests

Abstract

A new inequality between some functional of probability distribution functions is given. The inequality is based on strict convexity of a function used in functional definition. Equality sign in the inequality gives a characteristic property of some probability distributions. This fact together with special character of functional is used to construct free-of-distribution two sample tests. Key words: convex functions; probability distances; characterization of distributions; Cram\'{e}r - von Mises distance; statistical tests.