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June 2003 SPRT and CUSUM in hidden Markov models
Cheng-Der Fuh
Ann. Statist. 31(3): 942-977 (June 2003). DOI: 10.1214/aos/1056562468


In this paper, we study the problems of sequential probability ratio tests for parameterized hidden Markov models. We investigate in some detail the performance of the tests and derive corrected Brownian approximations for error probabilities and expected sample sizes. Asymptotic optimality of the sequential probability ratio test for testing simple hypotheses based on hidden Markov chain data is established. Next, we consider the cumulative sum (CUSUM) procedure for change point detection in this model. Based on the renewal property of the stopping rule, CUSUM can be regarded as a repeated one-sided sequential probability ratio test. Asymptotic optimality of the CUSUM procedure is proved in the sense of Lorden (1971). Motivated by the sequential analysis in hidden Markov models, Wald's likelihood ratio identity and Wald's equation for products of Markov random matrices are also given. We apply these results to several types of hidden Markov models: i.i.d. hidden Markov models, switch Gaussian regression and switch Gaussian autoregression, which are commonly used in digital communications, speech recognition, bioinformatics and economics.


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Cheng-Der Fuh. "SPRT and CUSUM in hidden Markov models." Ann. Statist. 31 (3) 942 - 977, June 2003.


Published: June 2003
First available in Project Euclid: 25 June 2003

zbMATH: 1036.60005
MathSciNet: MR1994736
Digital Object Identifier: 10.1214/aos/1056562468

Primary: 60B15
Secondary: 60F05 , 60K15

Keywords: Brownian approximation , change point detection , CUSUM , First passage time , Products of random matrices , renewal theory , Wald's equation , Wald's identity

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 3 • June 2003
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