Optimal error exponents in hidden Markov models order estimation

IEEE Transactions on Information Theory

Volumn 49, Issue 4, 2003, Pages 964-980

  Gassiat, Elisabeth ^a   Boucheron, Stéphane ^b  

^a UNIVERSITÉ PARIS SACLAY   (France)

^b CNRS   (France)

Author keywords

Composite hypothesis testing; Error exponents; Generalized likelihood ratio testing; Hidden Markov model (HMM); Large deviations; Order estimation; Stein's lemma

Indexed keywords

ERROR ANALYSIS; MAXIMUM LIKELIHOOD ESTIMATION; PROBABILITY DISTRIBUTIONS;

HIDDEN MARKOV MODELS (HMM);

MARKOV PROCESSES;

EID: 0037399347     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2003.809574     Document Type: Article

References (48)

1. L. E. Baum and T. Petrie, "Statistical inference for probabilistic functions of finite state Markov chains," Ann. Math. Statist., vol. 30, pp. 1554-1563, 1966.
2. A. Barron, J. Rissanen, and B. Yu., "The minimum description length principle in coding and modeling," IEEE Trans. Inform. Theory, vol. 44, pp. 2743-2760, Oct. 1998.
3. P. Bartlett, S. Boucheron, and G. Lugosi, "Model selection and error estimation," Machine Learning, vol. 48, pp. 85-113, 2002.
4. P. Bickel, Y. Ritov, and T. Ryden, "Asymptotic normality of the maximum likelihood estimator for general hidden Markov models," Ann. Statist., vol. 26, pp. 1614-1635, 1998.
5. O. Catoni, "Statistical learning and stochastic optimization," in Ecole d'été de Probabilities de Saint-Flour (Lecture Notes in Mathematics). Berlin, Germany: Springer-Verlag, 2001. [Online]. Available: http://felix.proba.jussieu.fr/users/catoni, to be published.
6. T. Cover and J. Thomas, Elements of Information Theory. New York: Wiley, 1991.
7. I. Csiszár, "Information theoretical methods in statistics," lecture notes, University of Maryland, College Park, 1990.
8. _, "The method of types," IEEE Trans. Inform. Theory, vol. 44, pp. 2505-2523, Oct. 1998.
9. _, "Large scale typicality of Markov sample paths and consistency of MDL order estimators," IEEE Trans. Inform. Theory (Special Issue on Shannon Theory: Perspective, Trends, and Applications), vol. 48, pp. 1616-1628, June 2002.
10. I. Csiszar and P. Shields, "The consistency of the BIC Markov order estimator," Ann. Statist., vol. 28, pp. 1601-1619, 2000.
11. L. A. Davisson, R. J. McEliece, M. B. Pursley, and M. S. Wallace, "Efficient universal noiseless source codes," IEEE Trans. Inform. Theory, vol. IT-27, pp. 269-279, May 1981.
12. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. New York: Springer-Verlag, 1999.
13. L. Devroye, L. Györfi, and G. Lugosi, A Probabilistic Theory of Pattern Recognition. Berlin, Germany: Springer-Verlag, 1996.
14. R. Doue and C. Matias, "Asymptotics of the maximum likelihood estimator for general hidden Markov models," Bernoulli, vol. 3, pp. 381-420, 2001.
15. R. Doue, E. Moulines, and T. Ryden, "Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime for general hidden Markov models," preprint, 2001.
16. P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations. New York: Wiley, 1997.
17. R. Dudley, Real Analysis and Probability. Belmont, CA: Wadsworth, 1989.
18. Y. Ephraim and N. Merhav, "Hidden Markov processes," IEEE Trans. Inform. Theory (Special Issue on Shannon Theory: Perspective, Trends, and Applications), vol. 48, pp. 1518-1569, June 2002.
19. M. Feder and N. Merhav, "Universal composite hypothesis testing: A competitive minimax approach. Special issue on Shannon theory: Perspective, trends, and applications," IEEE Trans. Inform. Theory (Special Issue on Shannon Theory: Perspective, Trends, and Applications), pp. 1504-1517, June 2002.
20. L. Finesso, "Estimation of the order of a finite Markov chain," Ph.D. dissertation, Univ. Maryland, College Park, 1990.
21. L. Finesso, C. Liu, and P. Narayan, "The optimal error exponent for Markov order estimation," IEEE Trans. Inform. Theory, vol. 42, pp. 1488-1497, Sept. 1996.
22. E. Gassiat and C. Keribin, "The likelihood ratio test for the number of components in a mixture with Markov regime," ESAIM Probab. Statist., vol. 4, pp. 25-52, 2000.
23. E. Gassiat, "Likelihood ratio inequalities with applications to various mixtures," Ann. Inst. Henri Poincaré, vol. 38, pp. 897-906, 2002.
24. W. Hoeffding, "Asymptotically optimal tests for multinomial distributions," Ann. Math. Statist., vol. 36, pp. 369-400, 1965.
25. H. Ito, S. I. Amari, and K. Kobayashi, "Identifiability of hidden Markov information sources and their minimum degrees of freedom," IEEE Trans. Inform. Theory, vol. 38, pp. 324-333, Mar. 1992.
26. J. L. Jensen and N. V. Petersen, "Asymptotic normality of the maximum likelihood estimator in state space models," Ann. Statist., vol. 27, pp. 514-535, 1999.
27. B. H. Juang and L. R. Rabiner, "Hidden Markov models for speech recognition," Technometrics, vol. 33, pp. 251-272, 1991.
28. J. C. Kieffer, "Strongly consistent code-based identification and order estimation for constrained finite state model classes," IEEE Trans. Inform. Theory, vol. 39, pp. 893-902, May 1993.
29. R. E. Krichevski and V. N. Trofimov, "The performance of universal encoding," IEEE Trans. Inform. Theory, vol. IT-27, pp. 199-207, Mar. 1981.
30. S. Khudanpur and P. Narayan, "Order estimation for a special class of hidden Markov sources and binary renewal processes," IEEE Trans. Inform. Theory (Special Issue on Shannon Theory: Perspective, Trends, and Applications), vol. 48, pp. 1704-1713, June 2002.
31. F. Le Gland and L. Mevel, "Exponential forgetting and geometric ergodicity in hidden Markov models," Math. Contr. Signals Syst., vol. 13, pp. 63-93, 2000.
32. B. Leroux, "Maximum-likelihood estimation for hidden Markov models," Stoch. Processing and Appl., vol. 40, pp. 127-143, 1992.
33. C. Liu and P. Narayan, "Order estimation and sequential universal data compression of a hidden Markov source by the method of mixtures," IEEE Trans. Inform. Theory, vol. 40, pp. 1167-1180, July 1994.
34. E. Levitan and N. Merhav, "A competitive Neyman-Pearson approach to universal hypothesis testing with applications," IEEE Trans. Inform. Theory, vol. 48, pp. 2215-2229, Aug. 2002.
35. I. L. MacDonald and W. Zucchini, Hidden Markov and Other Models for Discrete-Valued Time Series (Monographs on Statistics and Applied Probability), 70. London, U.K.: Chapman & Hall, 1997.
36. N. Merhav, M. Gutman, and J. Ziv, "On the estimation of the order of a Markov Chain and universal data compression," IEEE Trans. Inform. Theory, vol. 35, pp. 1014-1019, Sept. 1989.
37. T. Petrie, "Probabilistic functions of finite state Markov chains," Ann. Math. Statist, vol. 40, pp. 97-115, 1969.
38. J. Rissanen, "Universal coding, information, prediction and estimation," IEEE Trans. Inform. Theory, vol. IT-30, pp. 629-636, July 1984.
39. _, Stochastic Complexity in Statistical Enquiry. Singapore: World Scientific, 1989.
40. T. Rydén, "Estimating the order of a hidden Markov models," Statistics, vol. 26, pp. 345-354, 1995.
41. Y. M. Shtarkov, "Universal sequential coding of single messages," Probl. Inform. Transm., vol. 23, pp. 3-17, 1987.
42. A. van der Vaart, Asymptotic Statistics. Cambridge, U.K.: Cambridge Univ. Press, 1998.
43. V. Vapnik, Statistical Learning Theory. New York: Wiley, 1998.
44. F. M. J. Willems, Y. M. Shtarkov, and T. J. Tjalkens, "The context-tree weighting method: Basic properties," IEEE Trans. Inform. Theory, vol. 41, pp. 653-664, May 1995.
45. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. Cambridge, U.K.: Cambridge Math. Lib., 1924.
46. O. Zeitouni and M. Gutman, "On universal hypothesis testing via large deviations," IEEE Trans. Inform. Theory, vol. 37, pp. 285-290, Mar. 1991.
47. O. Zeitouni, J. Ziv, and N. Merhav, "When is the generalized likelihood ratio testing optimal?," IEEE Trans. Inform. Theory, vol. 38, pp. 1597-1602, Sept. 1992.
48. J. Ziv and N. Merhav, "Estimating the number of states of a finite-state source," IEEE Trans. Inform. Theory, vol. 38, pp. 61-65, Jan. 1992.