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October 2004 Asymptotic operating characteristics of an optimal change point detection in hidden Markov models
Cheng-Der Fuh
Ann. Statist. 32(5): 2305-2339 (October 2004). DOI: 10.1214/009053604000000580

Abstract

Let ξ0,ξ1,,ξω1 be observations from the hidden Markov model with probability distribution Pθ0, and let ξω,ξω+1, be observations from the hidden Markov model with probability distribution Pθ1. The parameters θ0 and θ1 are given, while the change point ω is unknown. The problem is to raise an alarm as soon as possible after the distribution changes from Pθ0 to Pθ1, but to avoid false alarms. Specifically, we seek a stopping rule N which allows us to observe the ξ's sequentially, such that EN is large, and subject to this constraint, sup kEk(Nk|Nk) is as small as possible. Here Ek denotes expectation under the change point k, and E denotes expectation under the hypothesis of no change whatever.

In this paper we investigate the performance of the Shiryayev–Roberts–Pollak (SRP) rule for change point detection in the dynamic system of hidden Markov models. By making use of Markov chain representation for the likelihood function, the structure of asymptotically minimax policy and of the Bayes rule, and sequential hypothesis testing theory for Markov random walks, we show that the SRP procedure is asymptotically minimax in the sense of Pollak [Ann. Statist. 13 (1985) 206–227]. Next, we present a second-order asymptotic approximation for the expected stopping time of such a stopping scheme when ω=1. Motivated by the sequential analysis in hidden Markov models, a nonlinear renewal theory for Markov random walks is also given.

Citation

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Cheng-Der Fuh. "Asymptotic operating characteristics of an optimal change point detection in hidden Markov models." Ann. Statist. 32 (5) 2305 - 2339, October 2004. https://doi.org/10.1214/009053604000000580

Information

Published: October 2004
First available in Project Euclid: 27 October 2004

zbMATH: 1073.60047
MathSciNet: MR2102511
Digital Object Identifier: 10.1214/009053604000000580

Subjects:
Primary: 60B15
Secondary: 60F05 , 60K15

Keywords: asymptotic optimality , change point detection , First passage time , limit of Bayes rules , nonlinear Markov renewal theory , Products of random matrices , Shiryayev–Roberts–Pollak procedure

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 5 • October 2004
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