Optimal Rank-Based Tests for the Location Parameter of a Rotationally Symmetric Distribution on the Hypersphere

Rotationally symmetric distributions on the unit hyperpshere are among the most successful ones in directional statistics. These distributions involve a finite-dimensional parameter θ and an infinite-dimensional parameter g1, that play the role of “location” and “angular density” parameters, respectively. In this paper, we consider semiparametric inference on θ, under unspecified g1. More precisely, we focus on hypothesis testing and develop tests for null hypotheses of the form H0 : θ = θ0, for some fixed θ0. Using the uniform local and asymptotic normality result from Ley et al. (2013), we define parametric tests that achieve Le Cam optimality at a target angular density f1. To improve on the poor robustness of these parametric procedures, we then introduce a class of rank tests for the same problem. Parallel to parametric tests, the proposed rank tests achieve Le Cam optimality under correctly specified angular densi-