On optimality of the Shiryaev-Roberts procedure for detecting a change in distribution

TL;DR

This paper investigates the optimality of the Shiryaev-Roberts procedure for change detection, showing that the commonly used randomized version is not strictly optimal and proposing a deterministic variant that is.

Contribution

It provides a counterexample demonstrating the non-optimality of Pollak's randomized procedure and introduces a deterministic initial point version that is strictly optimal.

Findings

Pollak's procedure is not strictly minimax.

A deterministic initial point can achieve strict optimality.

The Shiryaev-Roberts procedure can be improved with a specific initial condition.

Abstract

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.