The Annals of Statistics

A note on optimal detection of a change in distribution

Benjamin Yakir

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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


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