Asymptotic Operating Characteristics of an Optimal Change Point Detection in Hidden Markov Models

Asymptotic Operating Characteristics of an Optimal Change Point Detection in Hidden Markov Models

Let ξ0,ξ1,...,ξω-1be 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 θ0and θ1are given, while the change point ω is unknown...

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Veröffentlicht in:The Annals of Statistics. - Institute of Mathematical Statistics. - 32(2004), 5, Seite 2305-2339
1. Verfasser: Fuh, Cheng-Der (VerfasserIn)
Format: Online-Aufsatz
Sprache:English
Veröffentlicht: 2004
Zugriff auf das übergeordnete Werk:The Annals of Statistics
Schlagworte:Asymptotic Optimality Change Point Detection First Passage Time Limit of Bayes Rules Products of Random Matrices Nonlinear Markov Renewal Theory Shiryayev-Roberts-Pollak Procedure Mathematics Applied sciences Behavioral sciences
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520 |a Let ξ0,ξ1,...,ξω-1be 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 θ0and θ1are 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 E∞N is large, and subject to this constraint, supkEk(N - k∣N ≥ k) is as small as possible. Here Ekdenotes 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. 
540 |a Copyright 2004 Institute of Mathematical Statistics 
650 4 |a Asymptotic Optimality 
650 4 |a Change Point Detection 
650 4 |a First Passage Time 
650 4 |a Limit of Bayes Rules 
650 4 |a Products of Random Matrices 
650 4 |a Nonlinear Markov Renewal Theory 
650 4 |a Shiryayev-Roberts-Pollak Procedure 
650 4 |a Mathematics  |x Pure mathematics  |x Probability theory  |x Stochastic models  |x Markov models 
650 4 |a Mathematics  |x Pure mathematics  |x Probability theory  |x Random variables  |x Stochastic processes  |x Markov processes  |x Markov chains 
650 4 |a Applied sciences  |x Engineering  |x Automotive engineering  |x Stopping power  |x Stopping distances 
650 4 |a Mathematics  |x Applied mathematics  |x Statistics  |x Random walk 
650 4 |a Applied sciences  |x Systems science  |x Systems theory  |x Dynamical systems  |x Ergodic theory 
650 4 |a Behavioral sciences  |x Psychology  |x Cognitive psychology  |x Cognitive processes  |x Decision making  |x Bayesian theories  |x Bayes rule 
650 4 |a Mathematics  |x Mathematical procedures  |x Approximation 
650 4 |a Mathematics  |x Pure mathematics  |x Probability theory  |x Probabilities  |x Transition probabilities 
650 4 |a Mathematics  |x Applied mathematics  |x Game theory  |x Minimax 
650 4 |a Mathematics  |x Pure mathematics  |x Probability theory  |x Random variables  |x Hidden Markov Models 
655 4 |a research-article 
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