Multivariate Analysis of Nonparametric Estimates of Large Correlation Matrices

Abstract

We study concentration in spectral norm of nonparametric estimates of correlation matrices. We work within the confine of a Gaussian copula model. Two nonparametric estimators of the correlation matrix, the sine transformations of the Kendall's tau and Spearman's rho correlation coefficient, are studied. Expected spectrum error bound is obtained for both the estimators. A general large deviation bound for the maximum spectral error of a collection of submatrices of a given dimension is also established. These results prove that when both the number of variables and sample size are large, the spectral error of the nonparametric estimators is of no greater order than that of the latent sample covariance matrix, at least when compared with some of the sharpest known error bounds for the later. As an application, we establish the minimax optimal convergence rate in the estimation of high-dimensional bandable correlation matrices via tapering off of these nonparametric estimators. An optimal convergence rate for sparse principal component analysis is also established as another example of possible applications of the main results.

Robust covariance and scatter matrix estimation under Huber’s contamination model

“…To this end, we can claim that the proposed estimator by matrix depth have two extra robustness properties besides its rate optimality: resistance to outliers and insensitivity to heavy-tailedness. In fact, there are many works in the literature on scatter matrix estimation for elliptical distributions, including [50,69] in classical settings and [77,35,33,76,29,36,34,53,78] in high dimensional settings. However, it still remains open whether these estimators can achieve the minimax rates of the -contamination models.…”

Generative Adversarial Nets for Robust Scatter Estimation: A Proper Scoring Rule Perspective

“…In this setting, one has observations X 1 , ..., X n iid ∼ (1 − )N (0, Σ) + Q in R p , and the goal is to estimate the covariance matrix Σ using contaminated data without any assumption on the contamination distribution Q. Even though many robust covariance matrix estimators have been proposed and analyzed in the literature [40,56,63,28,43,58], the problem of optimal covariance estimation under the contamination model has not been investigated until the recent work by [9]. It was shown in [9] that the minimax rate with respect to the squared operator norm Σ−Σ 2 op is p n ∨ 2 .…”

High-Dimensional Gaussian Copula Regression: Adaptive Estimation and Statistical Inference

“…Rank-based correlation matrix estimation has been studied in a number of settings, including the nonparanormal graphical model [1,17,39], high dimensional structured covariance/precision matrix estimation [17,18,39], and sparse PCA model [10,24]. In the present paper, we only consider Kendall's tau-based estimator.…”