Optimal R-estimation of a spherical location

Abstract

In this paper, we provide R-estimators of the location of a rotationally symmetric distribution on the unit sphere of $\mathbb{R}^{k}$. In order to do so we first prove the local asymptotic normality property of a sequence of rotationally symmetric models; this is a non-standard result due to the curved nature of the unit sphere. We then construct our estimators by adapting the Le Cam one-step methodology to spherical statistics and ranks. We show that they are asymptotically normal under any rotationally symmetric distribution and achieve the efficiency bound under a specific density. Their small sample behavior is studied via a Monte Carlo simulation and our methodology is illustrated on geological data.