The Annals of Statistics

Quickest detection with exponential penalty for delay

H. Vincent Poor

Full-text: Open access

Abstract

The problem of detecting a change in the probability distribution of a random sequence is considered. Stopping times are derived that optimize the tradeoff between detection delay and false alarms within two criteria. In both cases, the detection delay is penalized exponentially rather than linearly, as has been the case in previous formulations of this problem. The first of these two criteria is to minimize a worst-case measure of the exponential detection delay within a lower-bound constraint on the mean time between false alarms. Expressions for the performance of the optimal detection rule are also developed for this case. It is seen, for example, that the classical Page CUSUM test can be arbitrarily unfavorable relative to the optimal test under exponential delay penalty. The second criterion considered is a Bayesian one, in which the unknown change point is assumed to obey a geometric prior distribution. In this case, the optimal stopping time effects an optimal trade-off between the expected exponential detection delay and the probability of false alarm. Finally, generalizations of these results to problems in which the penalties for delay may be path dependent are also considered.

Article information

Source
Ann. Statist. Volume 26, Number 6 (1998), 2179-2205.

Dates
First available in Project Euclid: 21 June 2002

Permanent link to this document
http://projecteuclid.org/euclid.aos/1024691466

Mathematical Reviews number (MathSciNet)
MR1700227

Digital Object Identifier
doi:10.1214/aos/1024691466

Zentralblatt MATH identifier
0927.62077

Subjects
Primary: 62L10: Sequential analysis
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 62L15 94A13: Detection theory

Keywords
Quickest detection change point problems optimal stopping exponential cost

Citation

Poor, H. Vincent. Quickest detection with exponential penalty for delay. The Annals of Statistics 26 (1998), no. 6, 2179--2205. doi:10.1214/aos/1024691466. http://projecteuclid.org/euclid.aos/1024691466.


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  • PRINCETON, NEW JERSEY 08544 E-MAIL: poor@princeton.edu