🤖 AI Summary
本文提出一种基于自举校准的谱散度检验方法，用于在线检测协方差矩阵的变化，解决了现有方法无法同时控制时间和窗口选择中误报率的问题。
📝 Abstract
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 $\widehatΣ_{0}^{-1/2}\widehatΣ_{t}\widehatΣ_{0}^{-1/2}$. 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.