Adaptive estimation of the transition density of a particular hidden Markov chain

Journal of Multivariate Analysis

Volumn 99, Issue 5, 2008, Pages 787-814

  Lacour, Claire ^a  

^a UNIVERSITÉ PARIS DESCARTES   (France)

Author keywords

Deconvolution; Hidden Markov chain; Model selection; Nonparametric estimation; Rate of convergence; Transition density

Indexed keywords

EID: 40949127508     PISSN: 0047259X     EISSN: 10957243     Source Type: Journal    
DOI: 10.1016/j.jmva.2007.04.006     Document Type: Article

References (32)

1. Bakry D., Milhaud X., and Vandekerkhove P. Statistique de chai{dotless}̂nes de Markov cachées à espace d'états fini. Le cas non stationnaire. C. R. Acad. Sci. Paris Sér. I. Math. 325 2 (1997) 203-206
2. Barron A., Birgé L., and Massart P. Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 3 (1999) 301-413
3. Baum L.E., and Petrie T. Statistical inference for probabilistic functions of finite state Markov chains. Ann. Math. Statist. 37 (1966) 1554-1563
4. Bickel P.J., Ritov Y., and Rydén T. Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models. Ann. Statist. 26 4 (1998) 1614-1635
5. Birgé L., and Massart P. Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli 4 3 (1998) 329-375
6. Butucea C. Deconvolution of supersmooth densities with smooth noise. Canad. J. Statist. 32 2 (2004) 181-192
7. Butucea C., and Matias C. Minimax estimation of the noise level and of the deconvolution density in a semiparametric convolution model. Bernoulli 11 2 (2005) 309-340
8. C. Butucea, A.B. Tsybakov, Sharp optimality for density deconvolution with dominating bias, Theory Probab. Appl. 52 (1) (2007), to appear.
9. Cappé O., Moulines E., and Rydén T. Inference in hidden Markov models, Springer Series in Statistics (2005), Springer, New York
10. Carroll R.J., and Hall P. Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 404 (1988) 1184-1186
11. Chaleyat-Maurel M., and Genon-Catalot V. Computable infinite dimensional filters with applications to discretized diffusions. Stochastic Process. Appl. 116 10 (2006) 1447-1467
12. {ring operator} 03-9; 2003, 〈http://www.u-paris10.fr/65897276/0/fiche_pagelibre/〉.
13. {ring operator} 2006-4; 2006a,〈http://www.math-info.univ-paris5.fr/map5/publis/titres06.html 〉.
14. Comte F., Rozenholc Y., and Taupin M.-L. Penalized contrast estimator for adaptive density deconvolution. Canad. J. Statist. 34 3 (2006) 431-452
15. Dorea C.C.Y., and Zhao L.C. Nonparametric density estimation in hidden Markov models. Statist. Inference Stochastic. Process. 5 1 (2002) 55-64
16. Douc R., Moulines E., and Rydén T. Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. Ann. Statist. 32 5 (2004) 2254-2304
17. P. Doukhan, Mixing. Properties and Examples, Lecture Notes in Statistics, vol. 85, Springer, New York, 1994.
18. Fan J. On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 3 (1991) 1257-1272
19. Genon-Catalot V., Jeantheau T., and Larédo C. Stochastic volatility models as hidden Markov models and statistical applications. Bernoulli 6 6 (2000) 1051-1079
20. Jensen J.L., and Petersen N.V. Asymptotic normality of the maximum likelihood estimator in state space models. Ann. Statist. 27 2 (1999) 514-535
21. Klein T., and Rio E. Concentration around the mean for maxima of empirical processes. Ann. Probab. 33 3 (2005) 1060-1077
22. Lacour C. Rates of convergence for nonparametric deconvolution. C. R. Acad. Sci. Paris Sér. I Math. 342 11 (2006) 877-882
23. C. Lacour, Adaptive estimation of the transition density of a Markov chain, Ann. Inst. H. Poincaré Probab. Statist, 2007, available online, to appear.
24. Lepski O.V., and Levit B.Y. Adaptive nonparametric estimation of smooth multivariate functions. Math. Methods Statist. 8 3 (1999) 344-370
25. Leroux B.G. Maximum-likelihood estimation for hidden Markov models. Stochastic Process. Appl. 40 1 (1992) 127-143
26. Masry E. Strong consistency and rates for deconvolution of multivariate densities of stationary processes. Stochastic Process. Appl. 47 1 (1993) 53-74
27. Matias C. Semiparametric deconvolution with unknown noise variance. ESAIM Probab. Statist. 6 (2002) 271-292 (electronic), new directions in time series analysis (Luminy, 2001)
28. Pensky M., and Vidakovic B. Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27 6 (1999) 2033-2053
29. Stefanski L.A. Rates of convergence of some estimators in a class of deconvolution problems. Statist. Probab. Lett. 9 3 (1990) 229-235
30. p adaptive density estimation in a β mixing framework. Ann. Inst. H. Poincaré Probab. Statist. 34 2 (1998) 179-208
31. Van Es B., Spreij P., and van Zanten H. Nonparametric volatility density estimation for discrete time models. J. Nonparametric Statist. 17 2 (2005) 237-251
32. Viennet G. Inequalities for absolutely regular sequences: application to density estimation. Probab. Theory Related Fields 107 4 (1997) 467-492