Improved Rates of Bootstrap Approximation for the Operator Norm: A CoordinateFree Approach
Abstract
Let $\hat\Sigma=\frac{1}{n}\sum_{i=1}^n X_i\otimes X_i$ denote the sample covariance operator of centered i.i.d. observations $X_1,\dots,X_n$ in a real separable Hilbert space, and let $\Sigma=\mathbf{E}(X_1\otimes X_1)$. The focus of this paper is to understand how well the bootstrap can approximate the distribution of the operator norm error $\sqrt n\\hat\Sigma\Sigma\_{\text{op}}$, in settings where the eigenvalues of $\Sigma$ decay as $\lambda_j(\Sigma)\asymp j^{2\beta}$ for some fixed parameter $\beta>1/2$. Our main result shows that the bootstrap can approximate the distribution of $\sqrt n\\hat\Sigma\Sigma\_{\text{op}}$ at a rate of order $n^{\frac{\beta1/2}{2\beta+4+\epsilon}}$ with respect to the Kolmogorov metric, for any fixed $\epsilon>0$. In particular, this shows that the bootstrap can achieve near $n^{1/2}$ rates in the regime of large $\beta$which substantially improves on previous near $n^{1/6}$ rates in the same regime. In addition to obtaining faster rates, our analysis leverages a fundamentally different perspective based on coordinatefree techniques. Moreover, our result holds in greater generality, and we propose a new model that is compatible with both elliptical and MarčenkoPastur models, which may be of independent interest.
 Publication:

arXiv eprints
 Pub Date:
 August 2022
 arXiv:
 arXiv:2208.03050
 Bibcode:
 2022arXiv220803050L
 Keywords:

 Mathematics  Statistics Theory;
 Mathematics  Probability