Bias-Corrected Multiplier Bootstrap Inference for Spectral Edges of Large Covariance Matrices

Abstract

Inference for spectral edges of large covariance matrices is a fundamental problem in high-dimensional statistics. A major difficulty is that the largest non-spiked sample eigenvalues, which serve as natural estimators of the edge, fluctuate on the Tracy--Widom scale. Consequently, valid inference requires accurate centering by the deterministic spectral edge together with a precise scaling constant, both of which are often difficult to estimate in practice under general unknown population covariance structures. In this paper, we propose a bias-corrected multiplier bootstrap procedure for inference on the deterministic edge of the bulk spectrum. The key idea is to introduce a carefully calibrated multiplier perturbation that regularizes the edge fluctuation to a slightly larger scale at which Gaussian approximation becomes tractable. The resulting confidence interval is constructed directly from bootstrap eigenvalues, together with a data-driven recentering step that corrects the bootstrap-induced shift of the deterministic edge. On the theoretical side, we show that, after bias correction and rescaling, the largest few non-spiked bootstrap eigenvalues are asymptotically Gaussian conditionally on the data. Building on this result, we establish the asymptotic validity of the proposed confidence interval, whose length is only slightly larger than the Tracy--Widom scale, and prove vanishing coverage under alternatives in which additional spikes separate from the bulk at a local scale larger than $n^{-1/6}$. As a consequence, the same confidence interval yields a threshold-free estimator for the number of spikes, without requiring the spikes to be distinct or very large. Equivalently, the procedure yields a data-driven and theoretically justified cutoff for the scree plot.