An Accurate Method for Determining the Pre-Change Run-Length Distribution of the Generalized Shiryaev--Roberts Detection Procedure

Change-of-measure is a powerful technique used in statistics, probability and analysis. Known as Wald's likelihood ratio identity, the technique enabled the proof of a number of exact and asymptotic optimality results pertaining to the problem of quickest change-point detection. Within the latter problem's context, we apply the technique to develop a numerical method to compute the Generalized Shiryaev--Roberts (GSR) detection procedure's pre-change Run-Length distribution and its various characteristics. Specifically, the numerical method is based on the integral-equation approach and exploits a martingale property of the GSR detection statistic. This along with the use of a change-of-measure ploy lend the method high accuracy. A tight bound on the method's error is supplied. The method is not restricted to a particular data distribution or to a specific value of the GSR detection statistic's headstart. To conclude, we offer a case study to demonstrate the proposed method at work, drawing particular attention to its accuracy including the rate of convergence. Specifically, assuming independent standard Gaussian observations undergoing a shift in the mean, we employ the method to study the GSR procedure's Run-Length's pre-change distribution, its average (i.e., the usual Average Run Length to false alarm) and standard deviation. We also comment on extending the method to other performance measures and procedures.