Optimal tests for circular reflective symmetry about an unknown central direction

TL;DR

This paper develops optimal parametric and semiparametric tests for circular reflective symmetry around an unknown center, demonstrating their effectiveness through simulations and real data applications.

Contribution

It introduces locally and asymptotically optimal tests for circular symmetry against asymmetric alternatives, with practical recommendations based on Monte Carlo studies.

Findings

Tests outperform previous methods in small- to medium-sized samples

Monte Carlo simulations validate the optimality of the proposed tests

Applications to real data demonstrate the methodology's usefulness

Abstract

Parametric and semiparametric tests of circular reflective symmetry about an unknown central direction are developed that are locally and asymptotically optimal in the Le Cam sense against asymmetric $k$-sine-skewed alternatives. The results from Monte Carlo studies comparing the rejection rates of tests with those of previously proposed tests lead to recommendations regarding the use of the various tests with small- to medium-sized samples. Analyses of data on the directions of cracks in cemented femoral components and the times of gun crimes in Pittsburgh illustrate the proposed methodology and its bootstrap extension.