Multivariate Analysis of Nonparametric Estimates of Large Correlation Matrices

TL;DR

This paper analyzes the spectral norm concentration of nonparametric correlation matrix estimators within a Gaussian copula model, providing bounds and optimal convergence rates for high-dimensional settings.

Contribution

It introduces spectral error bounds for sine transformations of Kendall's tau and Spearman's rho estimators, and establishes their optimal convergence rates in high-dimensional correlation matrix estimation.

Findings

Spectral error bounds match those of latent sample covariance matrices.

Established minimax optimal convergence rates for high-dimensional bandable correlation matrices.

Provided applications to sparse PCA with optimal convergence results.

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.