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 the analysis of massive data, however, the bootstrap is computationally intractable in its basic sampling-with-replacement version. Moreover, it is even not valid in some important high-dimensional applications. Combining subsampling of observations with suitable selection of their coordinates, we introduce a new ``$(m,mp/n)$ out of $(n,p)$''-sampling 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 is shown to consistently reproduce 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.
Personal website of Angelika Rode