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Change-of-measure is a powerful technique used across statistics, probabilityand analysis. Particularly known as Wald's likelihood ratio identity, thetechnique enabled the proof of a number of exact and asymptotic optimalityresults pertaining to the problem of quickest change-point detection. Withinthe latter problem's context we apply the technique to develop a numericalmethod to compute the Generalized Shiryaev--Roberts (GSR) detection procedure'spre-change Run-Length distribution. Specifically, the method is based on theintegral-equations approach and uses the collocation framework with the basisfunctions chosen so as to exploit a certain change-of-measure identity and aspecific martingale property of the GSR procedure's detection statistic. As aresult, the method's accuracy and robustness improve substantially, even thoughthe method's theoretical rate of convergence is shown to be merely quadratic. Atight upper bound on the method's error is supplied as well. The method is notrestricted to a particular data distribution or to a specific value of the GSRdetection statistic's "headstart". To conclude, we offer a case study todemonstrate the proposed method at work, drawing particular attention to themethod's accuracy and its robustness with respect to three factors: (a)partition size, (b) change magnitude, and (c) Average Run Length (ARL) to falsealarm level. Specifically, assuming independent standard Gaussian observationsundergoing a surge in the mean, we employ the method to study the GSRprocedure's Run-Length's pre-change distribution, its average (i.e., the usualARL to false alarm) and standard deviation. As expected from the theoreticalanalysis, the method's high accuracy and robustness with respect to theforegoing three factors are confirmed experimentally. We also comment onextending the method to handle other performance measures and other procedures.

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