CLT for Nonlinear Shrinkage Estimators of Large Covariance Matrices

Abstract

This thesis is concerned with finding the asymptotic distributions of linear spectral statistics of the nonlinear shrinkage estimator for large covariance matrices derived by Ledoit and Wolf (2012). It provides some new inferential procedures for large-dimensional data and shed some light on the power of the new statistical tests on the structure of the large covariance matrices. After a short review of the relevant theory, two linear spectral statistics are proposed which are deduced from the nonlinear shrinkage estimator where for one of these linear spectral statistics its limiting distribution is derived. This results in a ready to use sphericity test statistic in the large-dimensional framework and is one of the main results of this thesis. The sphericity test corresponding to this new test statistic is called the nonlinear shrinkage test (NLS-ε). The theoretical results are illustrated by means of a simulation study where the new nonlinear shrinkage test is compared with al- ready existing tests, in particular the commonly used corrected likelihood ratio test and the corrected John’s test. It is demonstrated that the new nonlinear shrinkage test is most powerful under a non homogeneous variance alternative where it outperforms well known sphericity tests. Moreover, it is observed that the new nonlinear shrinkage test encounters some problems when different alternatives are combined.