On Optimality of the Shiryaev-Roberts Procedure for Detecting Changes in Distributions

In 1985, for detecting changes in distributions 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. Pollak proved that this procedure is third-order asymptotically optimal as the mean time to false alarm becomes large. The question 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.