Tests for symmetry based on the integrated empirical process

Abstract

We propose simple Kolmogorov–Smirnov- and Cramér–von Mises-type tests for symmetry of a continuous distribution around an unknown center. The tests are based on a characterization involving symmetric integration of the distribution function around its median. The asymptotic null distributions of the test statistics are derived in the linear regression setup where the goal is to test for symmetry of the error distribution, and the tests are shown to be asymptotically consistent against general alternatives. Numerical results show that the tests are level preserving and have competitive power when compared to existing classical and modern tests for symmetry. Moreover, for the considered alternatives, the new tests seem to have consistently higher empirical power than the corresponding classical Kolmogorov–Smirnov and Cramér–von Mises tests.

Keywords:

• Characterization of symmetry
• goodness-of-fit test
• linear regression model
• test for symmetry

Acknowledgments

The authors would like to thank the Editor and a Reviewer for constructive remarks that led to a significantly improved version of the article.

Additional information

Funding

This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Number 114630).