2009/04/30 by Aleksey S. Polunchenko, Alexander G. Tartakovsky · 86 citations
Decision Sciences · Mathematics · #Advanced Statistical Methods and Models #Advanced Statistical Process Monitoring #Applied mathematics #Asymptotically optimal algorithm #Counterexample #Discrete mathematics #Mathematical optimization #Mathematics #Metric (unit) #Minimax #Random variable #Statistic #Statistical Distribution Estimation and Applications #Statistics #math.ST #stat.TH
paper · pdf · doi:10.1214/09-aos775
published in The Annals of Statistics 38(6) (Institute of Mathematical Statistics) · Published in at http://dx.doi.org/10.1214/09-AOS775 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2010/11/30 · arxiv created 2012/11/12 · arxiv updated 2015/03/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
In 1985, for detecting a change in distribution, Pollak introduced a specific minimax performance metric and a randomized version of the Shiryaev–Roberts procedure 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. In this paper, 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.