Spectrum estimation for large dimensional covariance matrices using random matrix theory

Abstract

Estimating the eigenvalues of a population covariance matrix from a sample covariance matrix is a problem of fundamental importance in multivariate statistics; the eigenvalues of covariance matrices play a key role in many widely used techniques, in particular in principal component analysis (PCA). In many modern data analysis problems, statisticians are faced with large datasets where the sample size, n, is of the same order of magnitude as the number of variables p. Random matrix theory predicts that in this context, the eigenvalues of the sample covariance matrix are not good estimators of the eigenvalues of the population covariance.

We propose to use a fundamental result in random matrix theory, the Marčenko–Pastur equation, to better estimate the eigenvalues of large dimensional covariance matrices. The Marčenko–Pastur equation holds in very wide generality and under weak assumptions. The estimator we obtain can be thought of as “shrinking” in a nonlinear fashion the eigenvalues of the sample covariance matrix to estimate the population eigenvalues. Inspired by ideas of random matrix theory, we also suggest a change of point of view when thinking about estimation of high-dimensional vectors: we do not try to estimate directly the vectors but rather a probability measure that describes them. We think this is a theoretically more fruitful way to think about these problems.

Our estimator gives fast and good or very good results in extended simulations. Our algorithmic approach is based on convex optimization. We also show that the proposed estimator is consistent.