Efficient performance evaluation of the generalized Shiryaev-Roberts detection procedure in a multi-cyclic setup

We propose a numerical method to evaluate the performance of the emerging Generalized Shiryaev-Roberts (GSR) change-point detection procedure in a 'minimax-ish' multi-cyclic setup where the procedure of choice is applied repetitively (cyclically), and the change is assumed to take place at an unknown time moment in a distant-future stationary regime. Specifically, the proposed method is based on the integral-equations approach and uses the collocation technique with the basis functions chosen so as to exploit a certain change-of-measure identity and the GSR detection statistic's unique martingale property. As a result, the method's accuracy and robustness improve, as does its efficiency as using the change-of-measure ploy the Average Run Length (ARL) to false alarm and the Stationary Average Detection Delay (STADD) are computed simultaneously. We show that the method's rate of convergence is quadratic and supply a tight upper bound on its error. We conclude with a case study and confirm experimentally that the proposed method's accuracy and rate of convergence are robust with respect to three factors: (a) partition fineness (coarse vs. fine), (b) change magnitude (faint vs. contrast), and (c) the level of the Average Run Length to false alarm (low vs. high). Because the method is designed not restricted to a particular data distribution or to a specific value of the GSR detection statistic's head start, this work may help gain greater insight into the characteristics of the GSR procedure and aid a practitioner to design the GSR procedure as needed while fully utilizing its potential.