Abstract not found

This paper provides optimal testing procedures for the m-sample null hypothesis of Common Principal Components (CPC) under possibly non Gaussian and heterogenous elliptical densi-ties. We first establish, under very mild assumptions that do not require finite moments of order four, the local asymptotic normality (LAN) of the model. Based on that result, we show that the pseudo-Gaussian test proposed in Hallin et al. (2010a) is locally and asymptotically optimal under Gaussian densities. We also show how to compute its local powers and asymptotic relative efficiencies (AREs). A numerical evaluation of those AREs, however, reveals that, while remain-ing valid, this test is poorly efficient away from the Gaussian. Moreover, it still requires finite moments of order four. We therefore propose rank-based procedures that remain valid under any possibly heterogenousm-tuple of elliptical densities, irrespective of any moment assumptions—in elliptical families, indeed, principal components naturally can be based on the scatter matrices characterizing the density contours, hence do not require finite variances. Those rank-based tests are not only validity-robust in the sense that they survive arbitrary elliptical population ∗Académie Royale de Belgique, CentER, Tilburg University, and ECORE. Research supported by the Sonderforschungsbereich “Statistical modelling of nonlinear dynamic processes ” (SFB 823) of the

Abstract not found

Document de travail

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Document de travail

Rotationally symmetric distributions on the unit hyperpshere are among the most successful ones in directional statistics. These distributions involve a finite-dimensional parameter θ and an infinite-dimensional parameter g1, that play the role of “location ” and “angular density ” parameters, respectively. In this paper, we consider semiparametric inference on θ, under unspecified g1. More precisely, we focus on hypothesis testing and develop tests for null hypotheses of the form H0: θ = θ0, for some fixed θ0. Using the uniform local and asymptotic normality result from Ley et al. (2013), we define parametric tests that achieve Le Cam optimality at a target angular density f1. To improve on the poor robustness of these parametric procedures, we then introduce a class of rank tests for the same problem. Parallel to parametric tests, the proposed rank tests achieve Le Cam optimality under correctly specified angular densi-1 ties. We derive the asymptotic properties of the various tests and investigate their finite-sample behaviour in a Monte Carlo study.

We propose rank-based estimators of principal components, both in the onesample and, under the assumption of common principal components, in the m-sample cases. Those estimators are obtained via a rank-based version of Le Cam’s one-step method, combined with an estimation of cross-information quantities. Under arbitrary elliptical distributions with, in the m-sample case, possibly heterogeneous radial densities, those R-estimators remain root-n consistent and asymptotically normal, while achieving asymptotic efficiency under correctly specified densities. Contrary to their traditional counterparts computed from empirical covariances, they do not require any moment conditions. When based on Gaussian score functions, in the one-sample case, they moreover uniformly dominate their classical

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.

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.