The Annals of Statistics

Asymptotic operating characteristics of an optimal change point detection in hidden Markov models

Cheng-Der Fuh

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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.

Article information

Source
Ann. Statist., Volume 32, Number 5 (2004), 2305-2339.

Dates
First available in Project Euclid: 27 October 2004

Permanent link to this document
https://projecteuclid.org/euclid.aos/1098883790

Digital Object Identifier
doi:10.1214/009053604000000580

Mathematical Reviews number (MathSciNet)
MR2102511

Zentralblatt MATH identifier
1073.60047

Subjects
Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Secondary: 60F05: Central limit and other weak theorems 60K15: Markov renewal processes, semi-Markov processes

Keywords
Asymptotic optimality change point detection first passage time limit of Bayes rules products of random matrices nonlinear Markov renewal theory Shiryayev–Roberts–Pollak procedure

Citation

Fuh, Cheng-Der. Asymptotic operating characteristics of an optimal change point detection in hidden Markov models. Ann. Statist. 32 (2004), no. 5, 2305--2339. doi:10.1214/009053604000000580. https://projecteuclid.org/euclid.aos/1098883790


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