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 $B \rightarrow \infty$.
Publié le : 1985-03-14
Classification:
Quality control,
control charts,
stopping time,
limit of Bayes rules,
minimax rule,
sequential analysis,
62L10,
62N10,
62C10,
62F15,
60G40
@article{1176346587,
author = {Pollak, Moshe},
title = {Optimal Detection of a Change in Distribution},
journal = {Ann. Statist.},
volume = {13},
number = {1},
year = {1985},
pages = { 206-227},
language = {en},
url = {http://dml.mathdoc.fr/item/1176346587}
}
Pollak, Moshe. Optimal Detection of a Change in Distribution. Ann. Statist., Tome 13 (1985) no. 1, pp. 206-227. http://gdmltest.u-ga.fr/item/1176346587/