Multivariate Analysis of Nonparametric Estimates of Correlation Matrices

• Activity Number: 174
• Type: Contributed
• Date/Time: Monday, August 4, 2014 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Ritwik Mitra*+ and Cun-Hui Zhang
• Companies: Rutgers University and Rutgers University
• Keywords: Nonparametric ; Gaussian Copula ; Correlation matrix ; Spectral norm ; PCA ; Tapering

Abstract

We study concentration in spectral norm of nonparametric estimates of correlation matrices. We work within the confines of a gaussian copula model. Two nonparametric estimates of elements of the correlation matrix, derived via sin transformations of the Kendall's tau and Spearman's rho correlation coefficient, are studied. Expected spectrum error bound is obtained for both the estimates. In addition, a general large deviation bound for any s-dimensional submatrix of the error is also derived. These results prove that in a non sparse double asymptotic setup, the spectral error bound of the nonparametric estimates matches the sharpest known rate of convergence given by the "oracle" sample correlation matrix estimator. As an application, minimax optimal convergence rate for sparse principal component analysis is established. We also establish a minimax optimal convergence rate in estimation of high dimensional bandable correlation matrices via tapering off of these nonparametric estimates.