We construct new testing procedures for spherical and elliptical symmetry based on the characterization that a random vector $X$ with finite mean has a spherical distribution if and only if $\Ex[u^\top X | v^\top X] = 0$ holds for any two perpendicular vectors $u$ and $v$. Our test is based on the Kolmogorov-Smirnov statistic, and its rejection region is found via the spherically symmetric bootstrap. We show the consistency of the spherically symmetric bootstrap test using a general Donsker theorem which is of some independent interest. For the case of testing for elliptical symmetry, the Kolmogorov-Smirnov statistic has an asymptotic drift term due to the estimated location and scale parameters. Therefore, an additional standardization is required in the bootstrap procedure. In a simulation study, the size and the power properties of our tests are assessed for several distributions and the performance is compared to that of several competing procedures.
This paper discusses fluctuations of linear spectral statistics of high-dimensional sample covariance matrices when the underlying population follows an elliptical distribution. Such population often possesses high order correlations among their coordinates, which have great impact on the asymptotic behaviors of linear spectral statistics. Taking such kind of dependency into consideration, we establish a new central limit theorem for the linear spectral statistics in this paper for a class of elliptical populations. This general theoretical result has wide applications and, as an example, it i…
We develop tests for high-dimensional covariance matrices under a generalized elliptical model. Our tests are based on a central limit theorem (CLT) for linear spectral statistics of the sample covariance matrix based on self-normalized observations. For testing sphericity, our tests neither assume specific parametric distributions nor involve the kurtosis of data. More generally, we can test against any non-negative definite matrix that can even be not invertible. As an interesting application, we illustrate in empirical studies that our tests can be used to test uncorrelatedness among idiosy…
This paper investigates improved testing inferences under a general multivariate elliptical regression model. The model is very flexible in terms of the specification of the mean vector and the dispersion matrix, and of the choice of the error distribution. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal and Student-t distributions as special cases. We obtain Skovgaard's adjusted likelihood ratio statistics and Barndorff-Nielsen's adjusted signed likelihood ratio statistics and we conduct a simula…
Although the assumption of elliptical symmetry is quite common in multivariate analysis and widespread in a number of applications, the problem of testing the null hypothesis of ellipticity so far has not been addressed in a fully satisfactory way. Most of the literature in the area indeed addresses the null hypothesis of elliptical symmetry with specified location and actually addresses location rather than non-elliptical alternatives. In this paper, we are proposing new classes of testing procedures, both for specified and unspecified location. The backbone of our construction is Le Cam's as…
Elliptical Regularized Hotelling Testing for High Dimensional Data
School of Statistics and Data Science, Nankai University（南开大学统计与数据科学学院）；Department of Mathematics, Hong Kong Baptist University（香港浸会大学数学系）；Advanced Institute of Natural Sciences, Beijing Normal University（北京师范大学前沿交叉科学研究院）
Some ideas studied by Lele (1993), under a Gaussian perturbation model, are generalised in the setting of matrix multivariate elliptical distributions. In particular, several inaccuracies in the published statistical perturbation model are revised. In addition, a number of aspects about identifiability and estimability are also considered. Instead of using the Euclidean distance matrix for proposing consistent estimates, this paper determines exact formulae for the moments of matrix $\mathbf{B} = \mathbf{X}^{c}\left(\mathbf{X}^{c}\right)^{T}$, where $\mathbf{X}^{c}$ is the centered landmarks m…
Several matrix variate hypergeometric type distributions are derived. The compound distributions of left-spherical matrix variate elliptical distributions and inverted hypergeometric type distributions with matrix arguments are then proposed. The scale mixture of left-spherical matrix variate elliptical distributions and univariate inverted hypergeometric type distributions is also derived as a particular case of the compound distribution approach.
This survey is devoted to recent developments in the statistical analysis of spherical data, with a view to applications in Cosmology. We will start from a brief discussion of Cosmological questions and motivations, arguing that most Cosmological observables are spherical random fields. Then, we will introduce some mathematical background on spherical random fields, including spectral representations and the construction of needlet and wavelet frames. We will then focus on some specific issues, including tools and algorithms for map reconstruction (\textit{i.e.}, separating the different physi…
In this paper, we investigate first the asymptotics of the minima of elliptical triangular arrays. Motivated by the findings of Kabluchko (Extremes 14 (2011) 285-310), we discuss further the asymptotic behaviour of the maxima of elliptical triangular arrays with marginal distribution functions in the Gumbel or Weibull max-domain of attraction. We present an application concerning the asymptotics of the maximum and the minimum of independent spherical processes.
In this paper we study the asymptotic behavior of the angular bispectrum of spherical random fields. Here, the asymptotic theory is developed in the framework of fixed-radius fields, which are observed with increasing resolution as the sample size grows. The results we present are then exploited in a set of procedures aimed at testing non-Gaussianity; for these statistics, we are able to show convergence to functionals of standard Brownian motion under the null hypothesis. Analytic results are also presented on the behavior of the tests in the presence of a broad class of non-Gaussian alternat…
We study the spherical cap packing problem with a probabilistic approach. Such probabilistic considerations result in an asymptotic sharp universal uniform bound on the maximal inner product between any set of unit vectors and a stochastically independent uniformly distributed unit vector. When the set of unit vectors are themselves independently uniformly distributed, we further develop the extreme value distribution limit of the maximal inner product, which characterizes its uncertainty around the bound. As applications of the above asymptotic results, we derive (1) an asymptotic sharp unive…
