Bootstrap Confidence Sets for Spectral Projectors of Sample Covariance

Let X1, . . . ,Xn be i.i.d. sample in Rp with zero mean and the covariance matrix . The problem of recovering the projector onto an eigenspace of from these observations naturally arises in many applications. Recent technique from [9] helps to study the asymp- totic distribution of the distance in the Frobenius norm kPr - bP rk2 between the true projector Pr on the subspace of the rth eigenvalue and its empirical counterpart bP r in terms of the effective rank of . This paper offers a bootstrap procedure for building sharp confidence sets for the true projector Pr from the given data. This procedure does not rely on the asymptotic distribution of kPr - bP rk2 and its moments. It could be applied for small or moderate sample size n and large dimension p. The main result states the validity of the proposed procedure for finite samples with an explicit error bound for the er- ror of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high-dimensional spaces. Numeric results confirm a good performance of the method in realistic examples.