Generalized Stochastic Approximation of the Log-Likelihood Ratio for Robust Sequential Change-Point Detection

Abstract

Generalized stochastic approximation replaces the log-likelihood ratio of a sequential change-point problem by an affine function of a fixed feature dictionary, whose coefficients solve a normal system in the dictionary's moments under the pre- and post-change laws. The increment is a rescaled orthogonal projection of the bounded score, and the information it carries is bounded by the triangular discrimination between the laws. The dictionary decides what is detectable: a symmetry shared by both laws annihilates a whole parity at every order, and one further column raises the information by an explicit bordered-metric increment. A single efficiency factor, at most one, governs all three procedures the increment drives -- cumulative-sum, Shiryaev-Roberts and Shiryaev -- through false-alarm and delay bounds valid at every threshold. Estimated coefficients are consistent and asymptotically normal. A Monte-Carlo study measures the factor against the delay ratio it predicts, and locates the order at which the dictionary stops paying.

arXiv commentv4: scope narrowed to the theory. The benchmark study of v1-v3 is withdrawn: a scoring error counted a run with no alarm as a detection, inflating the proposed method only. Theory and proofs stand. The normal system is now normalised F = C_0+C_1, halving J(s) and doubling theta_0 -- units, not a correction of Theorem 2. Full account on page 1. 29 pages