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**1 - 3**of**3**### 2011/71 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 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 curve ..."

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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.

### 2012/49 Optimal Tests for the Two-Sample Spherical Location Problem

"... We tackle the classical two-sample spherical location problem for directional data by having recourse to the Le Cam methodology, habitually used in classical “linear ” multivariate analysis. More precisely we construct locally and asymptotically optimal (in the maximin sense) parametric tests, which ..."

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We tackle the classical two-sample spherical location problem for directional data by having recourse to the Le Cam methodology, habitually used in classical “linear ” multivariate analysis. More precisely we construct locally and asymptotically optimal (in the maximin sense) parametric tests, which we then turn into semi-parametric ones in two distinct ways. First, by using a studentization argument; this leads to so-called pseudo-FvML tests. Second, by resorting to the invariance principle; this leads to efficient rank-based tests. Within each construction, the semi-parametric tests inherit optimality under a given distribution (the FvML in the first case, any rotationally symmetric one in the second) from their parametric counterparts and also improve on the latter by being valid under the whole class of rotationally symmetric distributions. Asymptotic relative efficiencies are calculated and the finite-sample behavior of the proposed tests is investigated by means of a Monte Carlo simulation.