Estimation in hidden Markov models via efficient importance sampling. (English) Zbl 1127.62068

Summary: Given a sequence of observations from a discrete-time, finite-state hidden Markov model, we would like to estimate the sampling distribution of a statistic. The bootstrap method is employed to approximate the confidence regions of a multi-dimensional parameter. We propose an importance sampling formula for efficient simulation in this context. Our approach consists of constructing a locally asymptotically normal (LAN) family of probability distributions around the default resampling rule and then minimizing the asymptotic variance within the LAN family. The solution of this minimization problem characterizes the asymptotically optimal resampling scheme, which is given by a tilting formula. The implementation of the tilting formula is facilitated by solving a Poisson equation. A few numerical examples are given to demonstrate the efficiency of the proposed importance sampling scheme.


62M05 Markov processes: estimation; hidden Markov models
65C60 Computational problems in statistics (MSC2010)
62F40 Bootstrap, jackknife and other resampling methods
Full Text: DOI arXiv


[1] Albert, P.S. (1991). A two-state Markov mixture model for a time series of epileptic seizure counts., Biometrics 47 1371–1381.
[2] Balish, M., Albert, P.S. and Theodore, W.H. (1991). Seizure frequency in intractable partial epilepsy: A statistical analysis., Epilepsia 32 642–649.
[3] Ball, F. and Rice, J.A. (1992). Stochastic models for ion channels: Introduction and bibliography., Math. Biosci. 112 189–206. · Zbl 0767.92007 · doi:10.1016/0025-5564(92)90023-P
[4] Baum, L.E. and Petrie, T. (1966). Statistical inference for probabilistic functions of finite state Markov chains., Ann. Math. Statist. 37 1554–1563. · Zbl 0144.40902 · doi:10.1214/aoms/1177699147
[5] Bickel, P., Ritov, Y. and Rydén, T. (1998). Asymptotic normality of the maximum likelihood estimator for general hidden Markov models., Ann. Statist. 26 1614–1635. · Zbl 0932.62097 · doi:10.1214/aos/1024691255
[6] Billingsley, P. (1961). Statistical methods in Markov chains., Ann. Math. Statist. 32 12–40. · Zbl 0104.12802 · doi:10.1214/aoms/1177705136
[7] Cappé, O., Moulines, E. and Rydén, T. (2005)., Inference in Hidden Markov Models. New York: Springer-Verlag. · Zbl 1080.62065
[8] Davison, A.C. (1988). Discussion of the papers by Hinkley and DiCiccio and Romano., J. R. Statist. Soc. B 50 356–357. JSTOR:
[9] Do, K.A. and Hall, P. (1991). On importance sampling for the bootstrap., Biometrika 78 161–167. JSTOR: · doi:10.1093/biomet/78.1.161
[10] Elliott, R., Aggoun, L. and Moore, J. (1995)., Hidden Markov Models: Estimation and Control. New York: Springer-Verlag. · Zbl 0819.60045
[11] Fuh, C.D. and Hu, I. (2000). Asymptotically efficient strategies for a stochastic scheduling problem with order constraints., Ann. Statist. 28 1670–1695. · Zbl 1105.62365 · doi:10.1214/aos/1015957475
[12] Fuh, C.D. and Hu, I. (2004). Efficient importance sampling for events of moderate deviations with applications., Biometrika 91 471–490. · Zbl 1079.62046 · doi:10.1093/biomet/91.2.471
[13] Ghosh, J.K. (1994)., Higher Order Asymptotics . NSF-CBMS Regional Conference Series. Hayward, CA: Institute of Mathematical Staistics.
[14] Hall, P. (1992)., The Bootstrap and Edgeworth Expansion. New York: Springer-Verlag. · Zbl 0744.62026
[15] Johns, M.V. (1988). Importance sampling for bootstrap confidence intervals., J. Amer. Statist. Assoc. 83 709–714. JSTOR: · Zbl 0664.62045 · doi:10.2307/2289294
[16] Krogh, A., Brown, M., Mian, I.S., Sjolander, K. and Haussler, D. (1994). Hidden Markov models in computational biology: application to protein modeling., J. Mol. Biol. 235 1501–1531.
[17] Künsch, H.R. (2001). State space and hidden Markov models. In D.E. Barndorff-Nielsen, D.R. Cox and C. Klüppelberg (eds), Complex Stochastic Systems , pp. 109–173. Boca Raton: Chapman & Hall/CRC. · Zbl 1002.62072
[18] LeCam, L. and Yang, G.L. (2000)., Asymptotics in Statistics. New York: Springer-Verlag.
[19] Leroux, B.G. (1992). Maximum likelihood estimation for hidden Markov models., Stochastic Process. Appl. 40 127–143. · Zbl 0738.62081 · doi:10.1016/0304-4149(92)90141-C
[20] MacDonald, I.L. and Zucchini, W. (1997)., Hidden Markov and Other Models for Discrete-Valued Time Series. London: Chapman & Hall. · Zbl 0868.60036
[21] Miller, H. (1961). A convexity property in the theory of random variables on a finite Markov chain., Ann. Math. Statist. 32 1260–1270. · Zbl 0108.15101 · doi:10.1214/aoms/1177704865
[22] Meyn, S.P. and Tweedie, R.L. (1993)., Markov Chains and Stochastic Stability. London: Springer-Verlag. · Zbl 0925.60001
[23] Ney, P. and Nummelin, E. (1987). Markov additive processes I: Eigenvalue properties and limit theorems., Ann. Probab. 15 561–592. · Zbl 0625.60027 · doi:10.1214/aop/1176992159
[24] Rabiner, L.R. and Juang, B.H. (1993)., Fundamentals of Speech Recognition. Englewood Cliffs, NJ: Prentice Hall. · Zbl 0762.62036
[25] Stoffer, D.S. and Wall, D. (1991). Bootstrapping state space models: Gaussian maximum likelihood estimation and the Kalman filter., J. Amer. Statist. Assoc. 86 1024–1033. JSTOR: · Zbl 0850.62693 · doi:10.2307/2290521
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.