0
views
0
recommends
+1 Recommend
0 collections
    0
    shares
      • Record: found
      • Abstract: found
      • Article: found
      Is Open Access

      Rates of Bootstrap Approximation for Eigenvalues in High-Dimensional PCA

      Preprint
      ,

      Read this article at

      Bookmark
          There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

          Abstract

          In the context of principal components analysis (PCA), the bootstrap is commonly applied to solve a variety of inference problems, such as constructing confidence intervals for the eigenvalues of the population covariance matrix \(\Sigma\). However, when the data are high-dimensional, there are relatively few theoretical guarantees that quantify the performance of the bootstrap. Our aim in this paper is to analyze how well the bootstrap can approximate the joint distribution of the leading eigenvalues of the sample covariance matrix \(\hat\Sigma\), and we establish non-asymptotic rates of approximation with respect to the multivariate Kolmogorov metric. Under certain assumptions, we show that the bootstrap can achieve the dimension-free rate of \({\tt{r}}(\Sigma)/\sqrt n\) up to logarithmic factors, where \({\tt{r}}(\Sigma)\) is the effective rank of \(\Sigma\), and \(n\) is the sample size. From a methodological standpoint, our work also illustrates that applying a transformation to the eigenvalues of \(\hat\Sigma\) before bootstrapping is an important consideration in high-dimensional settings.

          Related collections

          Author and article information

          Journal
          15 April 2021
          Article
          2104.07328
          5b4012d3-f350-495e-8c6c-8b85f946dae3

          http://creativecommons.org/licenses/by/4.0/

          History
          Custom metadata
          49 pages
          math.ST stat.ME stat.TH

          Methodology,Statistics theory
          Methodology, Statistics theory

          Comments

          Comment on this article