## The Annals of Statistics

- Ann. Statist.
- Volume 25, Number 5 (1997), 2117-2126.

### A note on optimal detection of a change in distribution

#### Abstract

Suppose $X_1, X_2, \dots, X_{\nu - 1}$ are iid random variables with distribution $F_0$, and $X_{\nu}, X_{\nu + 1}, \dots$ are are iid with distributed $F_1$. The change point $\nu$ is unknown. The problem is to raise an alarm as soon as possible after the distribution changes from $F_0$ to $F_1$ (detect the change), but to avoid false alarms.

Pollak found a version of the Shiryayev-Roberts procedure to be asymptotically optimal for the problem of minimizing the average run length to detection over all stopping times which satisfy a given constraint on the rate of false alarms. Here we find that this procedure is strictly optimal for a slight reformulation of the problem he considered.

Explicit formulas are developed for the calculation of the average run length (both before and after the change) for the optimal stopping time.

#### Article information

**Source**

Ann. Statist., Volume 25, Number 5 (1997), 2117-2126.

**Dates**

First available in Project Euclid: 20 November 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1069362390

**Digital Object Identifier**

doi:10.1214/aos/1069362390

**Mathematical Reviews number (MathSciNet)**

MR1474086

**Zentralblatt MATH identifier**

0942.62088

**Subjects**

Primary: 62L10: Sequential analysis

Secondary: 62N10

**Keywords**

Quality control control charts minimax rule Bayes rule

#### Citation

Yakir, Benjamin. A note on optimal detection of a change in distribution. Ann. Statist. 25 (1997), no. 5, 2117--2126. doi:10.1214/aos/1069362390. https://projecteuclid.org/euclid.aos/1069362390