On optimality of the Shiryaev-Roberts procedure for detecting a change in distribution. (English) Zbl 1204.62141

Summary: For detecting a change in distribution, M. Pollak [Ann. Stat. 13, 206–227 (1985; Zbl 0573.62074)] introduced a specific minimax performance metric and a randomized version of the Shiryaev-Roberts procedure [A. N. Shiryaev, Theor. Probab. Appl. 8, 22–46 (1963); translation from Teor. Veroyatn. Primen. 8, 26–51 (1963; Zbl 0213.43804); S. W. Roberts, Technometrics 8, 411–430 (1966)] where the zero initial condition is replaced by a random variable sampled from the quasi-stationary distribution of the Shiryaev-Roberts statistic. Pollak proved that this procedure is third-order asymptotically optimal as the mean time to false alarm becomes large. The question of whether Pollak’s procedure is strictly minimax for any false alarm rate has been open for more than two decades, and there were several attempts to prove this strict optimality. We provide a counterexample which shows that Pollak’s procedure is not optimal and that there is a strictly optimal procedure which is nothing but the Shiryaev-Roberts procedure that starts with a specially designed deterministic point.


62L10 Sequential statistical analysis
62L15 Optimal stopping in statistics
60G40 Stopping times; optimal stopping problems; gambling theory
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