The Annals of Statistics

SPRT and CUSUM in hidden Markov models

Cheng-Der Fuh

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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.

Article information

Ann. Statist., Volume 31, Number 3 (2003), 942-977.

First available in Project Euclid: 25 June 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Brownian approximation change point detection CUSUM first passage time products of random matrices renewal theory Wald's identity Wald's equation


Fuh, Cheng-Der. SPRT and CUSUM in hidden Markov models. Ann. Statist. 31 (2003), no. 3, 942--977. doi:10.1214/aos/1056562468.

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