Bootstrap confidence sets for spectral projectors of sample covariance

• 成果类型: Article
• 署名作者: Naumov, Alexey; Spokoiny, Vladimir; Ulyanov, Vladimir
• 署名单位: HSE University (National Research University Higher School of Economics); Russian Academy of Sciences; Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; Lomonosov Moscow State University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-018-0877-2
• 发表日期: 2019
• 页码: 1091-1132

摘要

Let X1, ... ,X-n be i.i.d. sample in Rp with zero mean and the covariance matrix Sigma. The problem of recovering the projector onto an eigenspace of Sigma from these observations naturally arises in many applications. Recent technique from Koltchinskii and Lounici (Ann Stat 45(1):121-157,2017) helps to study the asymptotic distribution of the distance in the Frobenius norm parallel to P-r-(P) over cap (r)parallel to(2) between the true projector Pr on the subspace of the rth eigenvalue and its empirical counterpart (P) over cap (r) in terms of the effective rank of Sigma. 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 parallel to P-r-(P) over cap (r)parallel to(2) 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 Gaussian samples with an explicit error bound for the error 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.