Tests based on characterizations, and their efficiencies: a survey

TL;DR

This survey reviews recent characterization-based goodness-of-fit and symmetry tests, focusing on their statistical properties, efficiencies, and potential future research directions.

Contribution

It compiles and analyzes recent developments in characterization-based tests, including their limiting distributions, efficiencies, and introduces new research avenues.

Findings

New test statistics' limiting distributions are described.

Local Bahadur efficiencies are calculated and compared.

The survey highlights promising future research directions.

Abstract

A survey of goodness-of-fit and symmetry tests based on the characterization properties of distributions is presented. This approach became popular in recent years. In most cases the test statistics are functionals of $U$-empirical processes. The limiting distributions and large deviations of new statistics under the null hypothesis are described. Their local Bahadur efficiency for various parametric alternatives is calculated and compared with each other as well as with diverse previously known tests. We also describe new directions of possible research in this domain.