Bootstrap-Calibrated Spectral Divergence Tests for Online Detection of Covariance Matrix Changes

RESUMO

A covariance matrix rarely distorts in a single direction: a shift can expand all variances simultaneously, shift variance along one or two principal directions, displace spectral mass without changing marginal means, or rotate the dependence structure. Default mean-shift detectors are blind to these effects, and no existing online procedure calibrates a family of spectral deviation tests with provable false alarm control across both time and tracking window selection; this paper fills that gap. We consider four spectral deviations: $D_{KL}(P_{1}\|P_{0})$, $D_{KL}(P_{0}\|P_{1})$, Jeffreys, and Bhattacharyya, each evaluated on the eigenvalues of the empirical relative covariance operator $\widehat{\Sigma}_{0}^{-1/2}\widehat{\Sigma}_{t}\widehat{\Sigma}_{0}^{-1/2}$. Critical values come from a conditional parametric bootstrap that accounts for estimation uncertainty in both the past and tracking windows, an aspect asymptotic approaches typically overlook. The procedure controls the false alarm rate family-by-family over a predefined monitoring period and a range of candidate window sizes; when a single operational window is needed, a power-based criterion selects it. We prove consistency under constant alternatives via local spectral expansions with second-order sensitivity near the null. Simulations are conservative under the null and show detection power depends largely on spectral shape rather than magnitude: $D_{KL}(P_{1}\|P_{0})$ excels under global inflation, while Jeffreys and Bhattacharyya cover a broader range of alternatives. We illustrate the approach on three financial applications: European stock indices, Fama-French sector portfolios, and large-cap technology stocks.