Faculty of Mathematics
Collaborative Research Centre 701
Spectral Structures and Topological Methods in Mathematics

# Wednesday, January 11, 2017 - 14:15 in V3-201

## Bootstrap confidence sets for spectral projectors of sample covariance

A talk in the 'Bielefeld Stochastic Afternoon - Oberseminar Math. Statistik und Wahrscheinlichkeitstheorie' series by
Alexey Naumov
 Abstract: Let $X_1$, ... ,$X_n$ be i.i.d. sample in $R^p$ with zero mean and the covariance matrix $S$. The problem of recovering the projector onto the eigenspace of $S$ from these observations naturally arises in many applications. Recent technique from [Koltchinskii and Lounici, 2015b] helps to study the asymptotic distribution of the distance in the Frobenius norm between the true projector $P_r$ on the subspace of the $r-th$ eigenvalue and its empirical counterpart $\hat{P}_r$ in terms of the effective trace of $S$. This paper offers a bootstrap procedure for building sharp confidence sets for the true projector $P_r$ from the given data. This procedure does not rely on the asymptotic distribution of $|| P_r - \hat{P}_r ||_2$ and its moments, it applies 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 on the error of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high dimension. Numeric results confirm a nice performance of the method in realistic examples. These are the joint results with V.Spokoiny and V. Ulyanov.

Within the CRC this talk is associated to the project(s): A4