Toward Optimal Design of the Generalized Shiryaev -- Roberts Procedure for Quickest Change-Point Detection under Exponential Observations

We consider the basic quickest change-point detection problem with optimality understood in Pollak´s minimax sense. The topic of interest is optimal design of the emerging Generalized Shiryaev-Roberts (GSR) detection procedure. To optimize the GSR procedure, we exploit the fact that the GSR procedure provides a lower bound on Pollak´s minimax Supremum (conditional) Average Detection Delay (SADD). Specifically, we propose to optimize the GSR procedure by choosing its head start and detection threshold so as to bring the lower bound as far up as is possible within the set tolerable Average Run Length (ARL) to false alarm level. We then follow through with this idea and carry out a case study where, in a specific exponential scenario, we solve the respective lower bound-vs-ARL tradeoff numerically, and tabulate the obtained optimal head start, detection threshold, and the maximized lower bound. The study is extensive in that it considers changes of diverse magnitudes and a wide range of levels of the ARL to false alarm, the latter are computed exactly. The study aids gain further insight into the GSR procedure as well as into the still-unsolved question of what minimizes Pollak´s SADD for a given level of the ARL to false alarm. Also, the tabulated optimal head start-detection-threshold pairs might help an engineer to properly set up the GSR procedure.