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Supplement to “Universal asymptotics for highdimensional sign tests
, 2013
"... In a smalln largep hypothesis testing framework, most procedures in the literature require quite stringent distributional assumptions, and restrict to a specific scheme of (n, p)asymptotics. More precisely, multinormality is almost always assumed, and it is imposed, typically, that p/n → c, fo ..."
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In a smalln largep hypothesis testing framework, most procedures in the literature require quite stringent distributional assumptions, and restrict to a specific scheme of (n, p)asymptotics. More precisely, multinormality is almost always assumed, and it is imposed, typically, that p/n → c, for some c in some given convex set C ⊂ (0,∞). Such restrictions clearly jeopardize practical relevance of these procedures. In this paper, we consider several classical testing problems in multivariate analysis, directional statistics, and multivariate time series: the problem of testing uniformity on the unit sphere, the spherical location problem, the problem of testing that a process is white noise versus serial dependence, the problem of testing for multivariate independence, and the problem of testing for sphericity. In each case, we show that the natural sign tests enjoy nonparametric validity and are distributionfree in a “universal ” (n, p)asymptotics framework, where p may go to infinity in an arbitrary way as n does. Simulations confirm our asymptotic results. 1. Introduction. There
reflective symmetry about a known median direction
"... In this paper, we propose optimal tests for reflective circular symmetry about a fixed median direction. The distributions against which optimality is achieved are the socalled ksineskewed distributions of Umbach and Jammalamadaka (2009). We first show that sequences of ksineskewed models are l ..."
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In this paper, we propose optimal tests for reflective circular symmetry about a fixed median direction. The distributions against which optimality is achieved are the socalled ksineskewed distributions of Umbach and Jammalamadaka (2009). We first show that sequences of ksineskewed models are locally and asymptotically normal in the vicinity of reflective symmetry. Following the Le Cam methodology, we then construct optimal (in the maximin sense) parametric tests for reflective symmetry, which we render semiparametric by a studentization argument. These asymptotically distributionfree tests happen to be uniformly optimal (under any reference density) and are moreover of a very simple form. They furthermore exhibit nice small sample properties, as we show through a Monte Carlo simulation study. Our new tests also allow us to revisit the famous red wood ants data set of Jander (1957). We further show that one of the proposed parametric tests can as well serve as a test for uniformity against cardioid alternatives; this test coincides with the famous circular Rayleigh (1919) test for uniformity which is thus proved to be (also) optimal against cardioid alternatives. Moreover, our choice of ksineskewed alternatives, which are the circular analogues of the classical linear skewsymmetric distributions, permits us a Fisher singularity analysis à la Hallin and Ley (2012) with the result that only the prominent sineskewed von Mises distribution suffers from these inferential drawbacks. Finally, we conclude the paper by discussing the unspecified location case.
2012/49 Optimal Tests for the TwoSample Spherical Location Problem
"... We tackle the classical twosample 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 twosample 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 semiparametric ones in two distinct ways. First, by using a studentization argument; this leads to socalled pseudoFvML tests. Second, by resorting to the invariance principle; this leads to efficient rankbased tests. Within each construction, the semiparametric 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 finitesample behavior of the proposed tests is investigated by means of a Monte Carlo simulation.
distributions on the sphere
"... A classical characterization result, which can be traced back to Gauss, states that the maximum likelihood estimator (MLE) of the location parameter equals the sample mean for any possible univariate samples of any possible sizes n if and only if the samples are drawn from a Gaussian population. A s ..."
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A classical characterization result, which can be traced back to Gauss, states that the maximum likelihood estimator (MLE) of the location parameter equals the sample mean for any possible univariate samples of any possible sizes n if and only if the samples are drawn from a Gaussian population. A similar result, in the twodimensional case, is given in von Mises (1929) for the Fishervon MisesLangevin (FVML) distribution, the equivalent of the Gaussian law on the unit circle. Half a century later, Bingham and Mardia (1975) extend the result to FVML distributions on the unit sphere Sk−1: = {v ∈ Rk: v ′ v = 1}, k ≥ 2. In this paper, we present a general MLE characterization theorem for a large subclass of rotationally symmetric distributions on Sk−1, k ≥ 2, including the FVML distribution. AMS (2000) subject classification. Primary 62H05 62E10; secondary 60E05.