Bootstrap-Calibrated Spectral Divergence Tests for Online Detection of Covariance Matrix Changes
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: DKL(P1|P0), DKL(P0|P1), Jeffreys, and Bhattacharyya, each evaluated on the eigenvalues of the empirical relative covariance operator Σ̂0−1/2Σ̂tΣ̂0−1/2. 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: DKL(P1|P0) 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.
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BibTeX
@misc{cadirci2026bootstrapcalibrated,
title = {{Bootstrap-Calibrated Spectral Divergence Tests for Online Detection of Covariance Matrix Changes}},
author = {Mehmet Siddik Cadirci and Martin Singull},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14506},
eprint = {2609.14506},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14506},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bootstrap-Calibrated Spectral Divergence Tests for Online Detection of Covariance Matrix Changes
AU - Mehmet Siddik Cadirci
AU - Martin Singull
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14506
UR - https://arxiv.org/abs/2609.14506
AB - 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 Σ̂_0^−1/2Σ̂_tΣ̂_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.
ER -