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Statistica Sinica 23 (2013), 305-332

doi:http://dx.doi.org/10.5705/ss.2011.206





OPTIMAL $\bm
R$-ESTIMATION OF A SPHERICAL LOCATION


Christophe Ley$^{1}$, Yvik Swan$^{2}$, Baba Thiam$^{3}$ and Thomas Verdebout$^{3}$


$^{1}$Université libre de Bruxelles, $^{2}$Université du Luxembourg,
and $^{3}$Université Lille Nord de France


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.



Key words and phrases: Local asymptotic normality, rank-based methods, R-estimation, spherical statistics.

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