## The Annals of Statistics

- Ann. Statist.
- Volume 38, Number 6 (2010), 3445-3457.

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

Aleksey S. Polunchenko and Alexander G. Tartakovsky

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 38, Number 6 (2010), 3445-3457.

**Dates**

First available in Project Euclid: 30 November 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1291126963

**Digital Object Identifier**

doi:10.1214/09-AOS775

**Mathematical Reviews number (MathSciNet)**

MR2766858

**Zentralblatt MATH identifier**

1204.62141

**Subjects**

Primary: 62L10: Sequential analysis 62L15: Optimal stopping [See also 60G40, 91A60]

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

**Keywords**

Changepoint problems Shiryaev–Roberts procedures sequential detection

#### Citation

Polunchenko, Aleksey S.; Tartakovsky, Alexander G. On optimality of the Shiryaev–Roberts procedure for detecting a change in distribution. Ann. Statist. 38 (2010), no. 6, 3445--3457. doi:10.1214/09-AOS775. https://projecteuclid.org/euclid.aos/1291126963