Testing uniformity on high-dimensional spheres against monotone rotationally symmetric alternatives

We consider the problem of testing uniformity on high-dimensional unit spheres. We are primarily interested in non-null issues. We show that rotationally symmetric alternatives lead to two Local Asymptotic Normality (LAN) structures. The first one is for fixed modal location $\theta$ and allows to derive locally asymptotically most powerful tests under specified $\theta$. The second one, that addresses the Fisher-von Mises-Langevin (FvML) case, relates to the unspecified-$\theta$ problem and shows that the high-dimensional Rayleigh test is locally asymptotically most powerful invariant. Under mild assumptions, we derive the asymptotic non-null distribution of this test, which allows to extend away from the FvML case the asymptotic powers obtained there from Le Cam's third lemma. Throughout, we allow the dimension $p$ to go to infinity in an arbitrary way as a function of the sample size $n$. Some of our results also strengthen the local optimality properties of the Rayleigh test in low dimensions. We perform a Monte Carlo study to illustrate our asymptotic results. Finally, we treat an application related to testing for sphericity in high dimensions

Detecting the direction of a signal on high-dimensional spheres: Non-null and Le Cam optimality results

We consider one of the most important problems in directional statistics, namely the problem of testing the null hypothesis that the spike direction $\theta$ of a Fisher-von Mises-Langevin distribution on the $p$-dimensional unit hypersphere is equal to a given direction $\theta_0$. After a reduction through invariance arguments, we derive local asymptotic normality (LAN) results in a general high-dimensional framework where the dimension $p_n$ goes to infinity at an arbitrary rate with the sample size $n$, and where the concentration $\kappa_n$ behaves in a completely free way with $n$, which offers a spectrum of problems ranging from arbitrarily easy to arbitrarily challenging ones. We identify various asymptotic regimes, depending on the convergence/divergence properties of $(\kappa_n)$, that yield different contiguity rates and different limiting experiments. In each regime, we derive Le Cam optimal tests under specified $\kappa_n$ and we compute, from the Le Cam third lemma, asymptotic powers of the classical Watson test under contiguous alternatives. We further establish LAN results with respect to both spike direction and concentration, which allows us to discuss optimality also under unspecified $\kappa_n$. To investigate the non-null behavior of the Watson test outside the parametric framework above, we derive its local asymptotic powers through martingale CLTs in the broader, semiparametric, model of rotationally symmetric distributions. A Monte Carlo study shows that the finite-sample behaviors of the various tests remarkably agree with our asymptotic results.Comment: 47 pages, 4 figure

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page

Optimal R-Estimation of a Spherical Location

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.Comment: Accepted in Statistica Sinic

Efficient ANOVA for directional data

In this paper, we tackle the ANOVA problem for directional data. We apply the invariance principle to construct locally and asymptotically most stringent rank-based tests. Our semi-parametric tests improve on the optimal parametric tests by being valid under the whole class of rotationally symmetric distributions. Moreover, they keep the optimality property of the latter under a given m-tuple 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. We conclude by applying our findings to a real-data example involving geological data

Efficient ANOVA for directional data

In this paper we tackle the ANOVA problem for directional data (with particular emphasis on geological data) by having recourse to the Le Cam methodology usually reserved for linear multivariate analysis. We construct locally and asymptotically most stringent parametric tests for ANOVA for directional data within the class of rotationally symmetric distributions. We turn these parametric tests into semi-parametric ones by (i) using a studentization argument (which leads to what we call pseudo-FvML tests) and by (ii) resorting to the invariance principle (which leads to efficient rank-based tests). Within each construction the semi-parametric tests inherit optimality under a given distribution (the FvML distribution in the first case, any rotationally symmetric distribution in the second) from their parametric antecedents 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. We conclude by applying our findings on a real-data example involving geological data

Some topics in the analysis of spherical data.

This thesis is concerned with the statistical analysis of directions in 3 dimensions. An important reference is the book by Mardia (1972). At the time of publication of this book, the repertoire of spherical distributions used for modelling purposes was rather limited, and there was clearly a need to investigate other possibilities. In the last few years there has been some interest in the 8 parameter family of distributions mentioned by Mardia (1975), which is known as the Fisher-Bingham family. In Chapter 1 an outline of the thesis is given. The Fisher-Bingham family is discussed in Chapter 2, and an effective method for calculating the normalising constant is presented. Attention is then focussed on an interesting 6 parameter subfamily, and a simple rule is given for classifying the distributions in this subfamily according to type (unimodal, bimodal, ’closed curve'). Estimation and inference are then discussed, and the Chapter is concluded with a numerical example. In Chapter 3, the family of bimodal distributions presented in Wood (1982) is described. Other bimodal models are also mentioned briefly. The problem of simulating Fisher-Bingham distributions is considered in Chapter 4. Some inequalities are derived and then used to construct suitable envelopes so that an acceptance-rejection procedure can be used. In Chapter 5, the robust estimation of concentration for a Fisher distribution is considered, and L-estimators of the type suggested by Fisher (1982) are investigated. It is shown that the best of these estimators have desirable all-round properties. Indications are also given as to how these ideas can be adapted to other contexts. Possibilities for further research are mentioned in Chapter 6

Small sphere distributions and related topics in directional statistics

This dissertation consists of two related topics in the statistical analysis of directional data. The research conducted for the dissertation is motivated by advancing the statistical shape analysis to understand the variation of shape changes in 3D objects. The first part of the dissertation studies a parametric approach for multivariate directional data lying on a product of spheres. Two kinds of concentric unimodal-small subsphere distributions are introduced. The first kind coincides with a special case of the Fisher-Bingham distribution; the second is a novel adaption that independently models horizontal and vertical variations. In its multi-subsphere version, the second kind allows for correlation of horizontal variations over different subspheres. For both kinds, we provide new computationally feasible algorithms for simulation and estimation, and propose a large-sample test procedure for several sets of hypotheses. Working as models to fit the major modes of variation, the proposed distributions properly describe shape changes of skeletally-represented 3D objects due to rotation, twisting and bending. In particular, the multi-subsphere version of the second kind accounts for the underlying horizontal dependence appropriately. The second part is a proposal of hypothesis test that is applicable to the analysis of principal nested spheres (PNS). In PNS, determining which subsphere to fit, among the geodesic (great) subsphere and non-geodesic (small) subsphere, is an important issue and it is preferred to fit a great subsphere when there is no major direction of variation in the directional data. The proposed test utilizes the measure of multivariate kurtosis. The change of the multivariate kurtosis for rotationally symmetric distributions is investigated based on modality. The test statistic is developed by modifying the sample kurtosis. The asymptotic sampling distribution of the test statistic is also investigated. The proposed test is seen to work well in numerical studies with various data situations