A note on optimal detection of a change in distribution
The Annals of Statistics · 1997 · Volume 25 · Issue 5
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.
Keywords
62L10, 62N10, Bayes rule, control charts, minimax rule, quality control
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