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Regular version of the site

Confidence Sets for Spectral Projectors of Covariance Matrices

Doklady Mathematics. 2018. Vol. 98. No. 2. P. 511-514.

A sample X_1,...,X_n consisting of шndependent identically distributed vectors in Rp with zero mean and a covariance matrix \Sigma 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 non-asymptotic bounds for the Frobenius norm \|\hat P_r − P_􏰀r \|_2^2 of the distance between sample and true projectors and studied asymptotic behavior for large samples. More specifically, asymptotic confidence sets for the true projector \P_r 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 \P_r of the covariance matrix \Sigmna from given data. This approach does not use the asymptotical distribution of \|\hat P_r − P_􏰀r \|_2^2 and does not require the computation of its moment characteristics. The performance of the bootstrap approximation procedure is analyzed.