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.