Abstract: We consider asymptotic inferences for circular data based on empirical characteristic functions. More precisely, we provide tests for reflective symmetry of circular data based on the imaginary part of the empirical characteristic function. We show that the proposed tests have many attractive features including the property of being locally and asymptotically maximin in the Le Cam sense under sine-skewed alternatives in the specified mean direction case. To the best of our knowledge, this result provides the first instance of such an optimality property for empirical characteristic functions. For the unspecified mean direction case, we provide corrected versions of the original tests that retain nice asymptotic power properties. The results are illustrated using a well-known data set and are checked using Monte-Carlo simulations.

Key words and phrases: Characteristic function, directional statistics, reflective symmetry.