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In this paper, we propose optimal tests for circular reflective symmetryabout a fixed median direction. The distributions against which optimality isachieved are the so-called k-sine-skewed distributions of Umbach andJammalamadaka (2009). We first show that sequences of k-sine-skewed models arelocally and asymptotically normal in the vicinity of reflective symmetry.Following the Le Cam methodology, we then construct optimal (in the maximinsense) parametric tests for reflective symmetry, which we rendersemi-parametric by a studentization argument. These asymptoticallydistribution-free tests happen to be uniformly optimal (under any referencedensity) and are moreover of a very simple and intuitive form. They furthermoreexhibit nice small sample properties, as we show through a Monte Carlosimulation study. Our new tests also allow us to re-visit the famous red woodants data set of Jander (1957). We further show that one of the proposedparametric tests can as well serve as a test for uniformity against cardioidalternatives; this test coincides with the famous circular Rayleigh (1919) testfor uniformity which is thus proved to be (also) optimal against cardioidalternatives. Moreover, our choice of k-sine-skewed alternatives, which are thecircular analogues of the classical linear skew-symmetric distributions,permits us a Fisher singularity analysis \`a la Hallin and Ley (2012) with theresult that only the prominent sine-skewed von Mises distribution suffers fromthese inferential drawbacks. Finally, we conclude the paper by discussing theunspecified location case.

Simple, asymptotically distribution-free, optimal tests for circular reflective symmetry about a known median direction