Humboldt-Universität zu Berlin - High Dimensional Nonstationary Time Series

IRTG1792DP2018 024

Bootstrap Confidence Sets for Spectral Projectors of Sample Covariance

A. Naumov
V. Spokoiny
V. Ulyanovk

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
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.


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