Confidence Sets for Spectral Projectors of Covariance Matrices

Abstract

A sample X1,...,Xn consisting of independent identically distributed vectors in ℝp with zero mean and a covariance matrix Σ is considered. The recovery of spectral projectors of high-dimensional covariance matrices from a sample of observations is a key problem in statistics arising in numerous applications. In their 2015 work, V. Koltchinskii and K. Lounici obtained nonasymptotic bounds for the Frobenius norm $$\parallel {P_r} - {\hat P_r}{\parallel _2}$$ of the distance between sample and true projectors and studied asymptotic behavior for large samples. More specifically, asymptotic confidence sets for the true projector Pr were constructed assuming that the moment characteristics of the observations are known. This paper describes a bootstrap procedure for constructing confidence sets for the spectral projector Pr of the covariance matrix Σ from given data. This approach does not use the asymptotical distribution of $$\parallel {P_r} - {\hat P_r}{\parallel _2}$$ and does not require the computation of its moment characteristics. The performance of the bootstrap approximation procedure is analyzed.