Computationally tractable nonparametric bootstrap of high-dimensional sample covariance matrices
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| Format: | Preprint |
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2024
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| author | Dette, Holger Rohde, Angelika |
| author_facet | Dette, Holger Rohde, Angelika |
| contents | We introduce a new ``$(m,mp/n)$ out of $(n,p)$'' sampling-with-replace\-ment bootstrap for eigenvalue statistics of high-dimensional sample covariance matrices based on $n$ independent $p$-dimensional random vectors. As it only uses $q=\lfloor mp/n\rfloor $ coordinates of the observations in a subsample of size $m \ll n $ from the original data, it is computationally tractable for large scale data. In the high-dimensional scenario $p/n\rightarrow c\in (0,\infty)$, this fully nonparametric bootstrap is shown to consistently reproduce the empirical 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. Our proofs are conducted under minimal moment requirements and incorporate delicate results on non-centered quadratic forms, combinatorial trace moments estimates as well as a conditional bootstrap martingale CLT which may be of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_16849 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computationally tractable nonparametric bootstrap of high-dimensional sample covariance matrices Dette, Holger Rohde, Angelika Statistics Theory Probability 62G09, 60F05 We introduce a new ``$(m,mp/n)$ out of $(n,p)$'' sampling-with-replace\-ment bootstrap for eigenvalue statistics of high-dimensional sample covariance matrices based on $n$ independent $p$-dimensional random vectors. As it only uses $q=\lfloor mp/n\rfloor $ coordinates of the observations in a subsample of size $m \ll n $ from the original data, it is computationally tractable for large scale data. In the high-dimensional scenario $p/n\rightarrow c\in (0,\infty)$, this fully nonparametric bootstrap is shown to consistently reproduce the empirical 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. Our proofs are conducted under minimal moment requirements and incorporate delicate results on non-centered quadratic forms, combinatorial trace moments estimates as well as a conditional bootstrap martingale CLT which may be of independent interest. |
| title | Computationally tractable nonparametric bootstrap of high-dimensional sample covariance matrices |
| topic | Statistics Theory Probability 62G09, 60F05 |
| url | https://arxiv.org/abs/2406.16849 |