# New characterization based symmetry tests

## TL;DR

This paper introduces two new symmetry tests based on characterization by squares of linear statistics, exploring their asymptotic properties and efficiency, and demonstrating their effectiveness compared to existing tests.

## Contribution

The paper proposes novel integral and Kolmogorov type symmetry tests derived from characterization by squares, with detailed asymptotic analysis and efficiency evaluation.

## Key findings

- Tests are comparable to classical symmetry tests.
- Asymptotic properties are rigorously derived.
- Efficiency assessments show competitive performance.

## Abstract

Two new symmetry tests, of integral and Kolmogorov type, based on the characterization by squares of linear statistics are proposed. The test statistics are related to the family of degenerate U-statistics. Their asymptotic properties are explored. The maximal eigenvalue, needed for the derivation of their logarithmic tail behavior, was calculated or approximated using techniques from the theory of linear operators and the perturbation theory. The quality of the tests is assessed using the approximate Bahadur efficiency as well as the simulated powers. The tests are shown to be comparable with some recent and classical tests of symmetry.