On optimal tests for circular reflective symmetry about an unknown central direction

Symmetry is one of the most fundamental of dividing hypotheses, its rejection, or not, heavily influencing subsequent modeling strategies. In this paper, the authors construct tests for circular reflective symmetry about an unknown central direction that are asymptotically valid within a semi-parametric class of distributions and maintain certain parametric local and asymptotic optimality properties. The asymptotic distributions of the test statistics under the null hypothesis and under local alternatives are established, and a pre-existing omnibus test is identified as a special case of the proposed construction. The finite-sample properties of the semi-parametric tests are compared with those of other testing approaches in a simulation experiment, and recommendations made regarding testing for reflective symmetry in practice. Analyses of data on the directions of cracks in hip replacements illustrate the proposed methodology.