Optimal rank-based tests for the location parameter of a rotationally symmetric distribution on the hypersphere

Abstract

Rotationally symmetric distributions on the unit hyperpshere are among the most commonly met in directional statistics. These distributions involve a finite-dimensional parameter $\boldsymbol{\theta}$ and an infinite-dimensional parameter $g$, that play the role of “location” and “angular density” parameters, respectively. In this paper, we focus on hypothesis testing on $\boldsymbol{\theta}$ , under unspecified $g$. We consider (i) the problem of testing that $\boldsymbol{\theta}$ is equal to some given $\boldsymbol{\theta}_0$, and (ii) the problem of testing that $\boldsymbol{\theta}$ belongs to some given great “circle”. Using the uniform local and asymptotic normality result from Ley et al. (Statistica Sinica 23:305–333, 2013), we define parametric tests that achieve Le Cam optimality at a target angular density $f$. To improve on the poor robustness of these parametric procedures, we then introduce a class of rank tests for these problems. Parallel to parametric tests, the proposed rank tests achieve Le Cam optimality under correctly specified angular densities. We derive the asymptotic properties of the various tests and investigate their finite-sample behavior in a Monte Catrlo study.