B1269
Title: Bootstrap of high-dimensional sample covariance matrices
Authors: Angelika Rohde - University of Freiburg (Germany) [presenting]
Abstract: Bootstrapping is the classical approach for distributional approximation of estimators and test statistics when an asymptotic distribution contains unknown quantities or provides a poor approximation quality. For massive data analysis, however, the bootstrap is computationally intractable in its basic sampling-with-replacement version. Moreover, it needs to be validated in some important high-dimensional applications. Combining subsampling of observations with a suitable selection of their coordinates, a new $(m,mp/n)$ out of $(n,p)$-sampling is introduced with replacement bootstrap for eigenvalue statistics of high-dimensional sample covariance matrices based on $n$ independent $p$-dimensional random vectors. In the high-dimensional scenario $p/n\rightarrow c\in (0,\infty)$, this fully nonparametric bootstrap consistently reproduces the underlying spectral measure if $m/n\rightarrow 0$. If $m^2/n\rightarrow 0$, it approximates correctly the distribution of linear spectral statistics. The crucial component is a suitably defined representative subpopulation condition, which is shown to be verified in a large variety of situations. The proofs incorporate several delicate technical results which may be of independent interest.
