## The Annals of Statistics

- Ann. Statist.
- Volume 13, Number 1 (1985), 206-227.

### Optimal Detection of a Change in Distribution

#### Abstract

Suppose one is able to observe sequentially a series of independent observations $X_1, X_2, \cdots$ such that $X_1, X_2, \cdots, X_{\nu-1}$ are iid distributed according to a known distribution $F_0$ and $X_\nu, X_{\nu+1}, \cdots$ are iid distributed according to a known distribution $F_1$. Assume that $\nu$ is unknown and the problem is to raise an alarm as soon as possible after the distribution changes from $F_0$ to $F_1$. Formally, the problem is to find a stopping rule $N$ which in some sense minimizes $E(N - \nu\mid N\geq \nu)$ subject to a restriction $E(N\mid\nu = \infty) \geq B$. A stopping rule that is a limit of Bayes rules is first derived. Then an almost minimax rule is presented; i.e. a stopping rule $N^\ast$ is described which satisfies $E(N^\ast\mid\nu = \infty) = B$ for which \begin{equation*}\begin{split}\sup_{1\leq\nu < \infty}E(N^\ast - \nu\mid N^\ast \geq \nu) \\ - \inf_{\{\text{stopping rules} N|E(N| \nu=\infty)\geq B\}} \sup_{1\leq\nu < \infty}E(N - \nu \mid N \geq \nu) = o(1)\end{split}\end{equation*} where $o(1) \rightarrow 0$ as $B \rightarrow \infty$.

#### Article information

**Source**

Ann. Statist., Volume 13, Number 1 (1985), 206-227.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176346587

**Digital Object Identifier**

doi:10.1214/aos/1176346587

**Mathematical Reviews number (MathSciNet)**

MR773162

**Zentralblatt MATH identifier**

0573.62074

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62L10: Sequential analysis

Secondary: 62N10 62C10: Bayesian problems; characterization of Bayes procedures 62F15: Bayesian inference 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

**Keywords**

Quality control control charts stopping time limit of Bayes rules minimax rule sequential analysis

#### Citation

Pollak, Moshe. Optimal Detection of a Change in Distribution. Ann. Statist. 13 (1985), no. 1, 206--227. doi:10.1214/aos/1176346587. https://projecteuclid.org/euclid.aos/1176346587