Optimal R-Estimation of a Spherical Location

TL;DR

This paper develops efficient R-estimators for the location parameter of rotationally symmetric distributions on the sphere, using Le Cam's one-step method, with proven asymptotic properties and practical applications.

Contribution

It introduces a novel adaptation of Le Cam's one-step estimation method to spherical data, establishing asymptotic normality and efficiency bounds for the estimators.

Findings

Estimators are asymptotically normal under any rotationally symmetric distribution.

The estimators achieve the efficiency bound at specific densities.

Monte Carlo simulations confirm good small sample performance.

Abstract

In this paper, we provide $R$-estimators of the location of a rotationally symmetric distribution on the unit sphere of $\R^k$. 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.