Year: 2021
Type: article
Abstract: In the probabilistic framework of multivariate regular variation, the first order behavior of heavy-tailed random vectors above large radial thresholds is ruled by a homogeneous limit measure. For a high dimensional vector, a reasonable assumption is that the support of this measure is concentrated on a lower dimensional subspace, meaning that certain linear combinations of the components are much likelier to be large than others. Identifying this subspace and thus reducing the dimension will facilitate a refined statistical analysis. In this work we apply Principal Component Analysis (PCA) to a re-scaled version of radially thresholded observations. Within the statistical learning framework of empirical risk minimization, our main focus is to analyze the squared reconstruction error for the exceedances over large radial thresholds. We prove that the empirical risk converges to the true risk, uniformly over all projection subspaces. As a consequence, the best projection subspace is shown to converge in probability to the optimal one, in terms of the Hausdorff distance between their intersections with the unit sphere. In addition, if the exceedances are re-scaled to the unit ball, we obtain finite sample uniform guarantees to the reconstruction error pertaining to the estimated projection subspace. Numerical experiments illustrate the capability of the proposed framework to improve estimators of extreme value parameters.
Source: Electronic Journal of Statistics
DOI: 10.1214/21-ejs1803
Authors: Holger Drees, Anne Sabourin
Language: English
Topic: Probability and Risk Models
Field: Decision Sciences
Domain: Social Sciences
Open Access status: gold
Estimated APC paid: $675
OpenAlex ID: W2955264995