UC Davis

High-dimensional statistical inference has become a fundamental topic in modern statistics, particularly in settings where the number of features is comparable to or even exceeds the sample size. Within this context, elliptical distributions are widely used as a flexible alternative to the multivariate normal distribution. Although recent research has substantially expanded the theoretical foundations of elliptical models in high dimensions, important methodological gaps remain. This thesis addresses two of them, by developing a goodness-of-fit procedure for validating the elliptical assumption and a bootstrap-based inference framework for spectral statistics. The first part of this thesis introduces a novel goodness-of-fit test to verify the elliptical assumption in high dimensions. We propose a test statistic based on the asymptotic equivalence of two distinct kurtosis estimators. This method avoids the estimation of an inverse covariance matrix, which prevents classical goodness-of-fit tests for elliptical models from being applicable in high-dimensional settings. Importantly, the asymptotic validity of the proposed test statistic is established under a general setting that requires no structural assumptions on the population covariance matrix. The second part introduces a parametric bootstrap methodology specifically designed for spectral statistics in high-dimensional elliptical models. Recognizing that the nonparametric bootstrap can be inconsistent for spectral statistics in this regime, we propose a bootstrap procedure that explicitly incorporates the elliptical structure of the data. We prove that this method consistently approximates the limiting distributions of linear spectral statistics by correctly capturing the fluctuations of the sample eigenvalues using recent advancements in high-dimensional central limit theorems. Extensive empirical results demonstrate the effectiveness of the method.