Abstract: In this paper we propose optimal tests for circular reflective symmetry about a fixed median direction. The distributions against which optimality is achieved are the k-sine-skewed distributions of Umbach and Jammalamadaka (2009). We first show that sequences of k-sine-skewed models are locally and asymptotically normal in the vicinity of reflective symmetry. Following the Le Cam methodology, we construct optimal (in the maximin sense) parametric tests for reflective symmetry, which we render semi-parametric by a studentization argument. These asymptotically distribution-free tests happen to be uniformly optimal (under any reference density) and are moreover of a simple and intuitive form. They furthermore exhibit nice small sample properties, as we show through a Monte Carlo simulation study. Our new tests also allow us to re-visit the famous red wood ants data set of Jander (1957). The choice of k-sine-skewed alternatives, which are the circular analogues of the Azzalini-type linear skew-symmetric distributions, permits us a Fisher singularity analysis \`a la Hallin and Ley (2012) with the result that only the prominent sine-skewed von Mises distribution suffers from these inferential drawbacks. We conclude the paper by discussing the unspecified location case.

Key words and phrases: Circular statistics, Fisher information singularity, skewed distributions, tests for symmetry.