Eigenspaces of covariance matrices play an important role in statistical machine learning, arising in variety of modern algorithms. Quantitatively, it is convenient to describe the eigenspaces in terms of spectral projectors. This work focuses on hypothesis testing for the spectral projectors, both in one- and two-sample scenario. We present new tests, based on a specific matrix norm developed in order to utilize the structure of the spectral projectors. A new resampling technique of independent interest is introduced and analyzed: it serves as an alternative to the well-known multiplier bootstrap, significantly reducing computational complexity of bootstrap-based methods. We provide theoretical guarantees for the type-I error of our procedures, which remarkably improve the previously obtained results in the field. Moreover, we analyze power of our tests. Numerical experiments illustrate good performance of the proposed methods compared to previously developed ones.
This paper considers testing a covariance matrix $Σ$ in the high dimensional setting where the dimension $p$ can be comparable or much larger than the sample size $n$. The problem of testing the hypothesis $H_0:Σ=Σ_0$ for a given covariance matrix $Σ_0$ is studied from a minimax point of view. We first characterize the boundary that separates the testable region from the non-testable region by the Frobenius norm when the ratio between the dimension $p$ over the sample size $n$ is bounded. A test based on a $U$-statistic is introduced and is shown to be rate optimal over this asymptotic regime.…
In this paper, we study the problem of testing the mean vectors of high dimensional data in both one-sample and two-sample cases. The proposed testing procedures employ maximum-type statistics and the parametric bootstrap techniques to compute the critical values. Different from the existing tests that heavily rely on the structural conditions on the unknown covariance matrices, the proposed tests allow general covariance structures of the data and therefore enjoy wide scope of applicability in practice. To enhance powers of the tests against sparse alternatives, we further propose two-step pr…
Asymptotic distribution for the proportional covariance model under multivariate normal distributions is derived. To this end, the parametrization of the common covariance matrix by its Cholesky root is adopted. The derivations are made in three steps. First, the asymptotic distribution of the maximum likelihood estimators of the proportionality coefficients and the Cholesky inverse root of the common covariance matrix is derived by finding the information matrix and its inverse. Next, the asymptotic distributions for the case of the Cholesky root of the common covariance matrix and finally fo…
We propose an independence test for random variables valued into metric spaces by using a test statistic obtained from appropriately centering and rescaling the squared Hilbert-Schmidt norm of the usual empirical estimator of normalized cross-covariance operator. We then get asymptotic normality of this statistic under independence hypothesis, so leading to a new test for independence of functional random variables. A simulation study that allows to compare the proposed test to existing ones is provided.
Let $X_{1},\ldots,X_{n}$ be i.i.d. sample in $\mathbb{R}^{p}$ with zero mean and the covariance matrix $\mathbfΣ$. The problem of recovering the projector onto an eigenspace of $\mathbfΣ$ from these observations naturally arises in many applications. Recent technique from [Koltchinskii, Lounici, 2015] helps to study the asymptotic distribution of the distance in the Frobenius norm $\| \mathbf{P}_r - \widehat{\mathbf{P}}_r \|_{2}$ between the true projector $\mathbf{P}_r$ on the subspace of the $r$-th eigenvalue and its empirical counterpart $\widehat{\mathbf{P}}_r$ in terms of the effective ra…
The Behrens-Fisher problem is a well-known hypothesis testing problem in statistics concerning two-sample mean comparison. In this article, we confirm one conjecture in Eaton and Olshen (1972), which provides stochastic bounds for the multivariate Behrens-Fisher test statistic under the null hypothesis. We also extend their results on the stochastic ordering of random quotients to the arbitrary finite dimensional case. This work can also be seen as a generalization of Hsu (1938) that provided the bounds for the univariate Behrens-Fisher problem. The results obtained in this article can be used…
We develop a group of robust, nonparametric hypothesis tests which detect differences between the covariance operators of several populations of functional data. These tests, called FKWC tests, are based on functional data depth ranks. These tests work well even when the data is heavy tailed, which is shown both in simulation and theoretically. These tests offer several other benefits, they have a simple distribution under the null hypothesis, they are computationally cheap and they possess transformation invariance properties. We show that under general alternative hypotheses these tests are …
The question of testing for equality in distribution between two linear models, each consisting of sums of distinct discrete independent random variables with unequal numbers of observations, has emerged from the biological research. In this case, the computation of classical $χ^2$ statistics, which would not include all observations, results in loss of power, especially when sample sizes are small. Here, as an alternative that uses all data, the nonparametric maximum likelihood estimator for the distribution of sum of discrete and independent random variables, which we call the convolution st…
The goal of this study was to test the equality of two covariance matrices by using modified Pillai's trace statistics under a high-dimensional framework, i.e., the dimension and sample sizes go to infinity proportionally. In this paper, we introduce two modified Pillai's trace statistics and obtain their asymptotic distributions under the null hypothesis. The benefits of the proposed statistics include the following: (1) the sample size can be smaller than the dimensions; (2) the limiting distributions of the proposed statistics are universal; and (3) we do not restrict the structure of the p…
We observe a sample of $n$ independent $p$-dimensional Gaussian vectors with Toeplitz covariance matrix $ Σ= [σ_{|i-j|}]_{1 \leq i,j \leq p}$ and $σ_0=1$. We consider the problem of testing the hypothesis that $Σ$ is the identity matrix asymptotically when $n \to \infty$ and $p \to \infty$. We suppose that the covariances $σ_k$ decrease either polynomially ($\sum_{k \geq 1} k^{2α} σ^2_{k} \leq L$ for $ α>1/4$ and $L>0$) or exponentially ($\sum_{k \geq 1} e^{2Ak} σ^2_{k} \leq L$ for $ A,L>0$). We consider a test procedure based on a weighted U-statistic of order 2, with optimal weights chosen a…
The matrix-variate normal distribution is a popular model for high-dimensional transposable data because it decomposes the dependence structure of the random matrix into the Kronecker product of two covariance matrices: one for each of the row and column variables. We develop tests for assessing the form of the row (column) covariance matrix in high-dimensional settings while treating the column (row) dependence structure as a nuisance. Our tests are robust to normality departures provided that the Kronecker product dependence structure holds. In simulations, we observe that the proposed tests…
This paper deals with the problem of parameter estimation based on certain eigenspaces of the empirical covariance matrix of an observed multidimensional time series, in the case where the time series dimension and the observation window grow to infinity at the same pace. In the area of large random matrix theory, recent contributions studied the behavior of the extreme eigenvalues of a random matrix and their associated eigenspaces when this matrix is subject to a fixed-rank perturbation. The present work is concerned with the situation where the parameters to be estimated determine the eigen…
This paper considers testing the covariance matrices structure based on Wald's score test in large dimensional setting. The hypothesis $H_0: Σ=Σ_0 $ for a given matrix $Σ_0$, which covers the identity hypothesis test and sphericity hypothesis test as the special cases, is reviewed by the generalized CLT (Central Limit Theorem) for the linear spectral statistics of large dimensional sample covariance matrices from Jiang(2015) . The proposed tests can be applicable for large dimensional non-Gaussian variables in a wider range. Furthermore, the simulation study is provided to compare the proposed…
Sample covariance matrices are widely used in multivariate statistical analysis. The central limit theorems (CLT's) for linear spectral statistics of high-dimensional non-centered sample covariance matrices have received considerable attention in random matrix theory and have been applied to many high-dimensional statistical problems. However, known population mean vectors are assumed for non-centered sample covariance matrices, some of which even assume Gaussian-like moment conditions. In fact, there are still another two most frequently used sample covariance matrices: the MLE (by subtractin…
In this paper, we study the asymptotic behavior of the extreme eigenvalues and eigenvectors of the high dimensional spiked sample covariance matrices, in the supercritical case when a reliable detection of spikes is possible. Especially, we derive the joint distribution of the extreme eigenvalues and the generalized components of the associated eigenvectors, i.e., the projections of the eigenvectors onto arbitrary given direction, assuming that the dimension and sample size are comparably large. In general, the joint distribution is given in terms of linear combinations of finitely many Gaussi…
When considering two or more time series of functions or curves, for instance those derived from densely observed intraday stock price data of several companies, the empirical cross-covariance operator is of fundamental importance due to its role in functional lagged regression and exploratory data analysis. Despite its relevance, statistical procedures for measuring the significance of such estimators are undeveloped. We present methodology based on a functional central limit theorem for conducting statistical inference for the cross-covariance operator estimated between two stationary, weakl…
Zhang (2019) presented a general estimation approach based on the Gaussian distribution for general parametric models where the likelihood of the data is difficult to obtain or unknown, but the mean and variance-covariance matrix are known. Castilla and Zografos (2021) extended the method to density power divergence-based estimators, which are more robust than the likelihood-based Gaussian estimator against data contamination. In this paper we introduce the restricted minimum density power divergence Gaussian estimator (MDPDGE) and study its main asymptotic properties. Also, we examine it robu…
This paper develops a new statistical inference theory for the precision matrix of high-frequency data in a high-dimensional setting. The focus is not only on point estimation but also on interval estimation and hypothesis testing for entries of the precision matrix. To accomplish this purpose, we establish an abstract asymptotic theory for the weighted graphical Lasso and its de-biased version without specifying the form of the initial covariance estimator. We also extend the scope of the theory to the case that a known factor structure is present in the data. The developed theory is applied …
Consider a one-way analysis of covariance model. Suppose that the parameter of interest theta is a specified linear contrast of the expected responses, for a given value of the covariate. Also suppose that the inference of interest is a 1-alpha confidence interval for theta. The following two-stage procedure has been proposed to determine the form of the model. In Stage 1, we carry out an F test of the null hypothesis that the slopes are all zero against the alternative hypothesis that they are not all zero. If this null hypothesis is accepted then we assume that the slopes are all zero; other…
Random Fisher matrices arise naturally in multivariate statistical analysis and understanding the properties of its eigenvalues is of primary importance for many hypothesis testing problems like testing the equality between two multivariate population covariance matrices, or testing the independence between sub-groups of a multivariate random vector. This paper is concerned with the properties of a large-dimensional Fisher matrix when the dimension of the population is proportionally large compared to the sample size. Most of existing works on Fisher matrices deal with a particular Fisher matr…
The assumption of separability of the covariance operator for a random image or hypersurface can be of substantial use in applications, especially in situations where the accurate estimation of the full covariance structure is unfeasible, either for computational reasons, or due to a small sample size. However, inferential tools to verify this assumption are somewhat lacking in high-dimensional or functional {data analysis} settings, where this assumption is most relevant. We propose here to test separability by focusing on $K$-dimensional projections of the difference between the covariance o…
In this paper we consider some hypothesis tests within a family of Wishart distributions, where both the sample space and the parameter space are symmetric cones. For such testing problems, we first derive the joint density of the ordered eigenvalues of the generalized Wishart distribution and propose a test statistic analog to that of classical multivariate statistics for testing homoscedasticity of covariance matrix. In this generalization of Bartlett's test for equality of variances to hypotheses of real, complex, quaternion, Lorentz and octonion types of covariance structures.
In this paper, we establish the central limit theorem (CLT) for linear spectral statistics (LSS) of large-dimensional sample covariance matrix when the population covariance matrices are not uniformly bounded, which is a nontrivial extension of the Bai-Silverstein theorem (BST) (2004). The latter has strongly stimulated the development of high-dimensional statistics, especially the application of random matrix theory to statistics. However, the assumption of uniform boundedness of the population covariance matrices is found strongly limited to the applications of BST. The aim of this paper is …
In this paper, we propose two new tests for testing the equality of the covariance functions of several functional populations, namely a quasi GPF test and a quasi $F_{\max}$ test. The asymptotic random expressions of the two tests under the null hypothesis are derived. We show that the asymptotic null distribution of the quasi GPF test is a chi-squared-type mixture whose distribution can be well approximated by a simple scaled chi-squared distribution. We also adopt a random permutation method for approximating the null distributions of the quasi GPF and $F_{\max}$ tests. The random permutati…
How many samples are sufficient to guarantee that the eigenvectors and eigenvalues of the sample covariance matrix are close to those of the actual covariance matrix? For a wide family of distributions, including distributions with finite second moment and distributions supported in a centered Euclidean ball, we prove that the inner product between eigenvectors of the sample and actual covariance matrices decreases proportionally to the respective eigenvalue distance. Our findings imply non-asymptotic concentration bounds for eigenvectors, eigenspaces, and eigenvalues. They also provide condit…
Structure discovery in graphical models is the determination of the topology of a graph that encodes conditional independence properties of the joint distribution of all variables in the model. For some class of probability distributions, an edge between two variables is present if and only if the corresponding entry in the precision matrix is non-zero. For a finite sample estimate of the precision matrix, entries close to zero may be due to low sample effects, or due to an actual association between variables; these two cases are not readily distinguishable. %Fisher provided a hypothesis test…
Multiple hypothesis testing is a fundamental problem in high dimensional inference, with wide applications in many scientific fields. In genome-wide association studies, tens of thousands of tests are performed simultaneously to find if any SNPs are associated with some traits and those tests are correlated. When test statistics are correlated, false discovery control becomes very challenging under arbitrary dependence. In the current paper, we propose a novel method based on principal factor approximation, which successfully subtracts the common dependence and weakens significantly the correl…
Under a multinormal distribution with arbitrary unknown covariance matrix, the main purpose of this paper is to propose a framework to achieve the goal of reconciliation of Bayesian, frequentist and Fisherian paradigms for the problems of testing mean against restricted alternatives (closed convex cones). Combining Fisher's fiducial inference and Wald's decision theory via d-admissibility into an unified approach, the goal can then be achieved. To proceed, the tests constructed via the union-intersection principle are studied.
University College London（伦敦大学学院）；University of Cambridge（剑桥大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Institute of Statistical Mathematics（统计数学研究所）
We provide a unifying framework linking two classes of statistics used in two-sample and independence testing: on the one hand, the energy distances and distance covariances from the statistics literature; on the other, maximum mean discrepancies (MMD), that is, distances between embeddings of distributions to reproducing kernel Hilbert spaces (RKHS), as established in machine learning. In the case where the energy distance is computed with a semimetric of negative type, a positive definite kernel, termed distance kernel, may be defined such that the MMD corresponds exactly to the energy dista…
Estimation and hypothesis tests for the covariance matrix in high dimensions is a challenging problem as the traditional multivariate asymptotic theory is no longer valid. When the dimension is larger than or increasing with the sample size, standard likelihood based tests for the covariance matrix have poor performance. Existing high dimensional tests are either computationally expensive or have very weak control of type I error. In this paper, we propose a test procedure, CRAMP, for testing hypotheses involving one or more covariance matrices using random projections. Projecting the high dim…
In this paper we propose methodology for inference of binary-valued adjacency matrices from various measures of the strength of association between pairs of network nodes, or more generally pairs of variables. This strength of association can be quantified by sample covariance and correlation matrices, and more generally by test-statistics and hypothesis test p-values from arbitrary distributions. Community detection methods such as block modelling typically require binary-valued adjacency matrices as a starting point. Hence, a main motivation for the methodology we propose is to obtain binary…
Based on a generalized cosine measure between two symmetric matrices, we propose a general framework for one-sample and two-sample tests of covariance and correlation matrices. We also develop a set of associated permutation algorithms for some common one-sample tests, such as the tests of sphericity, identity and compound symmetry, and the $K$-sample tests of multivariate equality of covariance or correlation matrices. The proposed method is very flexible in the sense that it does not assume any underlying distributions and data generation models. Moreover, it allows data to have different ma…
An important aspect of modeling spatially-referenced data is appropriately specifying the covariance function of the random field. A practitioner working with spatial data is presented a number of choices regarding the structure of the dependence between observations. One of these choices is determining whether or not an isotropic covariance function is appropriate. Isotropy implies that spatial dependence does not depend on the direction of the spatial separation between sampling locations. Misspecification of isotropy properties (directional dependence) can lead to misleading inferences, e.g…
We provide improved differentially private algorithms for identity testing of high-dimensional distributions. Specifically, for $d$-dimensional Gaussian distributions with known covariance $Σ$, we can test whether the distribution comes from $\mathcal{N}(μ^*, Σ)$ for some fixed $μ^*$ or from some $\mathcal{N}(μ, Σ)$ with total variation distance at least $α$ from $\mathcal{N}(μ^*, Σ)$ with $(\varepsilon, 0)$-differential privacy, using only \[\tilde{O}\left(\frac{d^{1/2}}{α^2} + \frac{d^{1/3}}{α^{4/3} \cdot \varepsilon^{2/3}} + \frac{1}{α\cdot \varepsilon}\right)\] samples if the algorithm is …
We study the accuracy of estimating the covariance and the precision matrix of a $D$-variate sub-Gaussian distribution along a prescribed subspace or direction using the finite sample covariance. Our results show that the estimation accuracy depends almost exclusively on the components of the distribution that correspond to desired subspaces or directions. This is relevant and important for problems where the behavior of data along a lower-dimensional space is of specific interest, such as dimension reduction or structured regression problems. We also show that estimation of precision matrices…
In this paper, we propose a new test for the equality of several covariance functions for functional data. Its test statistic is taken as the supremum value of the sum of the squared differences between the estimated individual covariance functions and the pooled sample covariance function, hoping to obtain a more powerful test than some existing tests for the same testing problem. The asymptotic random expression of this test statistic under the null hypothesis is obtained. To approximate the null distribution of the proposed test statistic, we describe a parametric bootstrap method and a non…
In this paper, we generalize the metric-based permutation test for the equality of covariance operators proposed by Pigoli et al. (2014) to the case of multiple samples of functional data. To this end, the non-parametric combination methodology of Pesarin and Salmaso (2010) is used to combine all the pairwise comparisons between samples into a global test. Different combining functions and permutation strategies are reviewed and analyzed in detail. The resulting test allows to make inference on the equality of the covariance operators of multiple groups and, if there is evidence to reject the …
Estimating covariance matrices is a problem of fundamental importance in multivariate statistics. In practice it is increasingly frequent to work with data matrices $X$ of dimension $n\times p$, where $p$ and $n$ are both large. Results from random matrix theory show very clearly that in this setting, standard estimators like the sample covariance matrix perform in general very poorly. In this "large $n$, large $p$" setting, it is sometimes the case that practitioners are willing to assume that many elements of the population covariance matrix are equal to 0, and hence this matrix is sparse. W…
Many experiments record sequential trajectories where each trajectory consists of oscillations and fluctuations around zero. Such trajectories can be viewed as zero-mean functional data. When there are structural breaks (on the sequence of trajectories) in higher order moments, it is not always easy to spot these by mere visual inspection. Motivated by this challenging problem in brain signal analysis, we propose a detection and testing procedure to find the change point in functional covariance. The detection procedure is based on the cumulative sum statistics (CUSUM). The classical testing p…
