Edgeworth corrections for the spiked eigenvalues of non-Gaussian sample covariance matrices with applications

TL;DR

This paper extends Edgeworth correction techniques to non-Gaussian sample covariance matrices, improving the accuracy of eigenvalue inference and spike estimation in high-dimensional statistics.

Contribution

It develops first-order Edgeworth expansions for non-Gaussian spiked eigenvalues, enabling more precise confidence intervals and a new estimator for the number of spikes.

Findings

Proposed Edgeworth-based confidence intervals outperform existing methods.

New estimator accurately detects the number of spikes in non-Gaussian data.

Method demonstrates robustness and improved accuracy in simulations.

Abstract

Yang and Johnstone (2018) established an Edgeworth correction for the largest sample eigenvalue in a spiked covariance model under the assumption of Gaussian observations, leaving the extension to non-Gaussian settings as an open problem. In this paper, we address this issue by establishing first-order Edgeworth expansions for spiked eigenvalues in both single-spike and multi-spike scenarios with non-Gaussian data. Leveraging these expansions, we construct more accurate confidence intervals for the population spiked eigenvalues and propose a novel estimator for the number of spikes. Simulation studies demonstrate that our proposed methodology outperforms existing approaches in both robustness and accuracy across a wide range of settings, particularly in low-dimensional cases.