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Parametric and semiparametric tests of circular reflective symmetry about anunknown central direction are developed that are locally and asymptoticallyoptimal in the Le Cam sense against asymmetric $k$-sine-skewed alternatives.The results from Monte Carlo studies comparing the rejection rates of testswith those of previously proposed tests lead to recommendations regarding theuse of the various tests with small- to medium-sized samples. Analyses of dataon the directions of cracks in cemented femoral components and the times of guncrimes in Pittsburgh illustrate the proposed methodology and its bootstrapextension.

Optimal tests for circular reflective symmetry about an unknown central direction