Bootstrap Confidence Sets

Abstract

A sample \( X_{1},\ldots ,X_{n} \) consisting of independent identically distributed random vectors in \( \mathbb {R}^{p} \) with zero mean and covariance matrix \( \mathbf {\Sigma }\) 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. This chapter describes a bootstrap procedure for constructing confidence sets for the spectral projector \( \mathbf {P}_{r} \) related to rth eigenvalue of the covariance matrix \(\mathbf {\Sigma }\) from given data on the base of corresponding spectral projector \(\widehat{\mathbf {P}}_{r}\) of the sample covariance matrix \(\widehat{\mathbf {\Sigma }}\). This approach does not use the asymptotical distribution of \( \Vert \mathbf {P}_{r} - \widehat{\mathbf {P}}_{r} \Vert _{2} \) and does not require the computation of its moment characteristics. The performance of the bootstrap approximation procedure is analyzed.