A fundamental problem in statistics is estimating the shape matrix of an Elliptical distribution. This generalizes the familiar problem of Gaussian covariance estimation, for which the sample covariance achieves optimal estimation error. For Elliptical distributions, Tyler proposed a natural M-estimator and showed strong statistical properties in the asymptotic regime, independent of the underlying distribution. Numerical experiments show that this estimator performs very well, and that Tyler's iterative procedure converges quickly to the estimator. Franks and Moitra recently provided the first distribution-free error bounds in the finite sample setting, as well as the first rigorous convergence analysis of Tyler's iterative procedure. However, their results exceed the sample complexity of the Gaussian setting by a $\log^{2} d$ factor. We close this gap by proving optimal sample threshold and error bounds for Tyler's M-estimator for all Elliptical distributions, fully matching the Gaussian result. Moreover, we recover the algorithmic convergence even at this lower sample threshold. Our approach builds on the operator scaling connection of Franks and Moitra by introducing a novel pseudorandom condition, which we call $\infty$-expansion. We show that Elliptical distributions satisfy $\infty$-expansion at the optimal sample threshold, and then prove a novel scaling result for inputs satisfying this condition.
Estimating a high-dimensional sparse covariance matrix from a limited number of samples is a fundamental problem in contemporary data analysis. Most proposals to date, however, are not robust to outliers or heavy tails. Towards bridging this gap, in this work we consider estimating a sparse shape matrix from $n$ samples following a possibly heavy tailed elliptical distribution. We propose estimators based on thresholding either Tyler's M-estimator or its regularized variant. We derive bounds on the difference in spectral norm between our estimators and the shape matrix in the joint limit as th…
In many problems from multivariate analysis, the parameter of interest is a shape matrix, that is, a normalized version of the corresponding scatter or dispersion matrix. In this paper, we propose a depth concept for shape matrices that involves data points only through their directions from the center of the distribution. We use the terminology Tyler shape depth since the resulting estimator of shape, namely the deepest shape matrix, is the median-based counterpart of the M-estimator of shape of Tyler (1987). Beyond estimation, shape depth, like its Tyler antecedent, also allows hypothesis te…
Technion - Israel Institute of Technology（以色列理工学院）
Tyler's M-estimator is a well known procedure for robust and heavy-tailed covariance estimation. Tyler himself suggested an iterative fixed-point algorithm for computing his estimator however, it requires super-linear (in the size of the data) runtime per iteration, which maybe prohibitive in large scale. In this work we propose, to the best of our knowledge, the first Frank-Wolfe-based algorithms for computing Tyler's estimator. One variant uses standard Frank-Wolfe steps, the second also considers \textit{away-steps} (AFW), and the third is a \textit{geodesic} version of AFW (GAFW). AFW prov…
Principal component analysis (PCA) is one of the most popular dimension reduction methods. The usual PCA is known to be sensitive to the presence of outliers, and thus many robust PCA methods have been developed. Among them, the Tyler's M-estimator is shown to be the most robust scatter estimator under the elliptical distribution. However, when the underlying distribution is contaminated and deviates from ellipticity, Tyler's M-estimator might not work well. In this article, we apply the semiparametric theory to propose a robust semiparametric PCA. The merits of our proposal are twofold. First…
Estimating the shape of an elliptical distribution is a fundamental problem in statistics. One estimator for the shape matrix, Tyler's M-estimator, has been shown to have many appealing asymptotic properties. It performs well in numerical experiments and can be quickly computed in practice by a simple iterative procedure. Despite the many years the estimator has been studied in the statistics community, there was neither a tight non-asymptotic bound on the rate of the estimator nor a proof that the iterative procedure converges in polynomially many steps. Here we observe a surprising connectio…
High-Dimensional Regularization of the Spatial Sign Covariance Matrix for Robust Shape Estimation
Whiting School of Engineering, Johns Hopkins University（约翰斯·霍普金斯大学惠廷工程学院）；Department of Applied Mathematics and Statistics, Johns Hopkins University（约翰斯·霍普金斯大学应用数学与统计学系）；Department of Statistics & Data Science Institute, University of Chicago（芝加哥大学统计与数据科学研究所）
Matrix factor model has been growing popular in scientific fields such as econometrics, which serves as a two-way dimension reduction tool for matrix sequences. In this article, we for the first time propose the matrix elliptical factor model, which can better depict the possible heavy-tailed property of matrix-valued data especially in finance. Manifold Principle Component Analysis (MPCA) is for the first time introduced to estimate the row/column loading spaces. MPCA first performs Singular Value Decomposition (SVD)for each "local" matrix observation and then averages the local estimated spa…
Distributed Principal Component Analysis (PCA) has been studied to deal with the case when data are stored across multiple machines and communication cost or privacy concerns prohibit the computation of PCA in a central location. However, the sub-Gaussian assumption in the related literature is restrictive in real application where outliers or heavy-tailed data are common in areas such as finance and macroeconomic. In this article, we propose a distributed algorithm for estimating the principle eigenspaces without any moment constraint on the underlying distribution. We study the problem under…
In this work, we focus on some conditional extreme risk measures estimation for elliptical random vectors. In a previous paper, we proposed a methodology to approximate extreme quantiles, based on two extremal parameters. We thus propose some estimators for these parameters, and study their asymptotic properties in the case of heavy-tailed distributions. Thereafter, from these parameters, we construct extreme conditional quantiles estimators, and give their consistency properties. Using recent results on the asymptotic relationship between quantiles and other risk measures, we deduce estimator…
Statistical analysis of tensor-valued data has largely used the tensor-variate normal (TVN) distribution that may be inadequate when data comes from distributions with heavier or lighter tails. We study a general family of elliptically contoured (EC) tensor-variate distributions and derive its characterizations, moments, marginal and conditional distributions, and the EC Wishart distribution. We describe procedures for maximum likelihood estimation from data that are (1) uncorrelated draws from an EC distribution, (2) from a scale mixture of the TVN distribution, and (3) from an underlying but…
In this paper, we consider the spectrum sensing in cognitive radio networks when the impulsive noise appears. We propose a class of blind and robust detectors using M-estimators in eigenvalue based spectrum sensing method. The conventional eigenvalue based method uses statistics derived from the eigenvalues of sample covariance matrix(SCM) as testing statistics, which are inefficient and unstable in the impulsive noise environment. Instead of SCM, we can use M-estimators, which have good performance under both impulsive and non-impulsive noise. Among those M-estimators, We recommend the Tyler'…
Consistency for constrained maximum likelihood estimation and clustering based on mixtures of elliptically-symmetric distributions under general data generating processes
In this paper, we consider the problem of estimating the $p\times p$ scale matrix $Σ$ of a multivariate linear regression model $Y=X\,β+ \mathcal{E}\,$ when the distribution of the observed matrix $Y$ belongs to a large class of elliptically symmetric distributions. After deriving the canonical form $(Z^T U^T)^T$ of this model, any estimator $\hat{ Σ}$ of $Σ$ is assessed through the data-based loss tr$(S^{+}Σ\, (Σ^{-1}\hatΣ - I_p)^2 )\,$ where $S=U^T U$ is the sample covariance matrix and $S^{+}$ is its Moore-Penrose inverse. We provide alternative estimators to the usual estimators $a\,S$, wh…
In this paper, we derive the joint asymptotic distributions of functions of quantile estimators (the non-parametric sample quantile and the parametric location-scale quantile estimator) with functions of measure of dispersion estimators (the sample variance, sample mean absolute deviation, sample median absolute deviation) - assuming an underlying identically and independently distributed sample. We also discuss the conditions required by the use of such estimators. Further, we show that these results can be extended to any higher order absolute central sample moment as measure of dispersion. …
We study the estimation of the covariance matrix $Σ$ of a $p$-dimensional normal random vector based on $n$ independent observations corrupted by additive noise. Only a general nonparametric assumption is imposed on the distribution of the noise without any sparsity constraint on its covariance matrix. In this high-dimensional semiparametric deconvolution problem, we propose spectral thresholding estimators that are adaptive to the sparsity of $Σ$. We establish an oracle inequality for these estimators under model miss-specification and derive non-asymptotic minimax convergence rates that are …
The sub-Gaussian stable distribution is a heavy-tailed elliptically contoured law which has interesting applications in signal processing and financial mathematics. This work addresses the problem of feasible estimation of distributions. We present a method based on application of the method of moments to the empirical characteristic function. Further, we show almost sure convergence of our estimators, discover their limiting distribution and demonstrate their finite-sample performance.
