Estimation of the transition density of a Markov chain
Sart, Mathieu
Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014), p. 1028-1068 / Harvested from Numdam
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Nous présentons deux procédures pour estimer la densité de transition d'une chaîne de Markov homogène. Dans la première procédure, nous construisons un estimateur constant par morceaux sur une partition aléatoire bien choisie. Nous établissons des bornes de risque non-asymptotiques pour une perte de type Hellinger lorsque la racine carrée de la densité de transition appartient à un espace de Besov inhomogène dont l'indice de régularité peut être petit. Nous illustrons ces résultats par des simulations numériques. La deuxième procédure est d'intérêt théorique. Elle permet d'obtenir un théorème de sélection de modèle à partir duquel nous déduisons des vitesses de convergence sur des espaces de Besov inhomogènes anisotropes. Nous étudions finalement les vitesses qui peuvent être atteintes sous des hypothèses structurelles sur la densité de transition.

We present two data-driven procedures to estimate the transition density of an homogeneous Markov chain. The first yields a piecewise constant estimator on a suitable random partition. By using an Hellinger-type loss, we establish non-asymptotic risk bounds for our estimator when the square root of the transition density belongs to possibly inhomogeneous Besov spaces with possibly small regularity index. Some simulations are also provided. The second procedure is of theoretical interest and leads to a general model selection theorem from which we derive rates of convergence over a very wide range of possibly inhomogeneous and anisotropic Besov spaces. We also investigate the rates that can be achieved under structural assumptions on the transition density.

Publié le : 2014-01-01
DOI : https://doi.org/10.1214/13-AIHP551
Classification:  62M05,  62G05 
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     author = {Sart, Mathieu},
     title = {Estimation of the transition density of a Markov chain},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {50},
     year = {2014},
     pages = {1028-1068},
     doi = {10.1214/13-AIHP551},
     mrnumber = {3224298},
     zbl = {1298.62144},
     language = {en},
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Sart, Mathieu. Estimation of the transition density of a Markov chain. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) pp. 1028-1068. doi : 10.1214/13-AIHP551. http://gdmltest.u-ga.fr/item/AIHPB_2014__50_3_1028_0/

[1] N. Akakpo. Estimation adaptative par sélection de partitions en rectangles dyadiques. Ph.D. thesis, Univ. Paris Sud, 2009.

[2] N. Akakpo. Adaptation to anisotropy and inhomogeneity via dyadic piecewise polynomial selection. Math. Methods Statist. 21 (2012) 1-28. | MR 2901269

[3] N. Akakpo and C. Lacour. Inhomogeneous and anisotropic conditional density estimation from dependent data. Electron. J. Statist. 5 (2011) 1618-1653. | MR 2870146 | Zbl 1271.62060

[4] K. B. Athreya and G. S. Atuncar. Kernel estimation for real-valued Markov chains. Sankhyā 60 (1998) 1-17. | MR 1714774 | Zbl 0977.62093

[5] Y. Baraud. Estimator selection with respect to Hellinger-type risks. Probab. Theory Related Fields 151 (2011) 353-401. | MR 2834722 | Zbl pre05968717

[6] Y. Baraud and L. Birgé. Estimating the intensity of a random measure by histogram type estimators. Probab. Theory Related Fields 143 (2009) 239-284. | MR 2449129 | Zbl 1149.62019

[7] Y. Baraud and L. Birgé. Estimating composite functions by model selection. Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014) 285-314. | Numdam | MR 3161532 | Zbl 1281.62093

[8] A. K. Basu and D. K. Sahoo. On Berry-Esseen theorem for nonparametric density estimation in Markov sequences. Bull. Inform. Cybernet. 30 (1998) 25-39. | MR 1629735 | Zbl 0921.62039

[9] L. Birgé. Approximation dans les espaces métriques et théorie de l'estimation. Probab. Theory Related Fields 65 (1983) 181-237. | MR 722129 | Zbl 0506.62026

[10] L Birgé. Stabilité et instabilité du risque minimax pour des variables indépendantes équidistribuées. Ann. Inst. Henri Poincaré Probab. Stat. 20 (1984) 201-223. | Numdam | Zbl 0542.62018

[11] L. Birgé. Sur un théorème de minimax et son application aux tests. Probab. Math. Statist. 2 (1984) 259-282. | MR 764150 | Zbl 0571.62036

[12] L. Birgé. Model selection via testing: An alternative to (penalized) maximum likelihood estimators. Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 273-325. | Numdam | MR 2219712 | Zbl pre05024238

[13] L. Birgé. Model selection for Poisson processes. In Asymptotics: Particles, Processes and Inverse Problems 32-64. IMS Lecture Notes Monogr. Ser. 55. IMS, Beachwood, OH, 2007. | MR 2459930 | Zbl 1176.62082

[14] L. Birgé. Model selection for density estimation with 𝕃 2 -loss. Probab. Theory Related Fields 158 (2014) 533-574. | MR 3176358 | Zbl 1285.62037

[15] L. Birgé. Robust tests for model selection. In From Probability to Statistics and Back: High-Dimensional Models and Processes. A Festschrift in Honor of Jon Wellner 47-64. IMS Collections 9. IMS, Beachwood, OH, 2012. | MR 3186748

[16] G. Blanchard, C. Schäfer and Y. Rozenholc. Oracle Bounds and Exact Algorithm for Dyadic Classification Trees. Lecture Notes in Comput. Sci. 3120. Springer, Berlin, 2004. | MR 2177922 | Zbl 1078.62521

[17] R. C. Bradley. Basic properties of strong mixing conditions. A survey and some open questions. Probab. Surv. 2 (2005) 107-144. | MR 2178042 | Zbl 1189.60077

[18] S. Clémencon. Adaptive estimation of the transition density of a regular Markov chain. Math. Methods Statist. 9 (2000) 323-357. | MR 1827473 | Zbl 1008.62076

[19] F. Comte and Y. Rozenholc. Adaptive estimation of mean and volatility functions in (auto-)regressive models. Stochastic Process. Appl. 97 (2002) 111-145. | MR 1870963 | Zbl 1064.62046

[20] W. Dahmen, R. Devore and K. Scherer. Multi-dimensional spline approximation. SIAM J. Numer. Anal. 17 (1980) 380-402. | MR 581486 | Zbl 0437.41010

[21] R. DeVore and X. Yu. Degree of adaptive approximation. Math. Comput. 55 (1990) 625-635. | MR 1035930 | Zbl 0723.41015

[22] C. C. Y. Dorea. Strong consistency of kernel estimators for Markov transition densities. Bull. Braz. Math. Soc. (N.S.) 33 (2002) 409-418. | MR 1978836 | Zbl 1033.62035

[23] P. Doukhan. Mixing: Properties and Examples. Lecture Notes in Statistics 85. Springer, New York, 1994. | MR 1312160 | Zbl 0801.60027

[24] P. Doukhan and M. Ghindès. Estimation de la transition de probabilité d'une chaîne de Markov Doëblin-récurrente 15 (1983) 271-293. | MR 711186 | Zbl 0515.62037

[25] R. Hochmuth. Wavelet characterizations for anisotropic Besov spaces. Appl. Comput. Harmon. Anal. 12 (2002) 179-208. | MR 1884234 | Zbl 1003.42024

[26] A. Juditsky, O. Lepski and A. Tsybakov. Nonparametric estimation of composite functions. Ann. Statist. 37 (2009) 1360-1404. | MR 2509077 | Zbl 1160.62030

[27] C. Lacour. Adaptive estimation of the transition density of a Markov chain. Ann. Inst. Henri Poincaré Probab. Statist. 43 (2007) 571-597. | Numdam | MR 2347097 | Zbl 1125.62087

[28] C. Lacour. Nonparametric estimation of the stationary density and the transition density of a Markov chain. Stochastic Process. Appl. 118 (2008) 232-260. | MR 2376901 | Zbl 1129.62028

[29] C. Lacour. Erratum to “Nonparametric estimation of the stationary density and the transition density of a Markov chain” [Stochastic Process. Appl. 118 (2008) 232-260] []. Stochastic Process. Appl. 122 (2012) 2480-2485. | MR 2376901 | Zbl 1277.62106

[30] L. Le Cam. Convergence of estimates under dimensionality restrictions. Ann. Statist. 1 (1973) 38-53. | MR 334381 | Zbl 0255.62006

[31] L. Le Cam. On local and global properties in the theory of asymptotic normality of experiments. In Stochastic Processes and Related Topics (Proc. Summer Res. Inst. Statist. Inference for Stochastic Processes, Indiana Univ., Bloomington, Ind., 1974, Vol. 1; dedicated to Jerzy Neyman) 13-54. Academic Press, New York, 1975. | MR 395005 | Zbl 0389.62011

[32] P. Massart. Concentration Inequalities and Model Selection. Lecture Notes in Mathematics 1896. Springer, Berlin, 2003. | MR 2319879 | Zbl 1170.60006

[33] G. G. Roussas. Nonparametric estimation in Markov processes. Ann. Inst. Statist. Math. 21 (1969) 73-87. | MR 247722 | Zbl 0181.45804

[34] G. G. Roussas. Estimation of Transition Distribution Function and Its Quantiles in Markov Processes: Strong Consistency and Asymptotic Normality. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 335. Kluwer Acad. Publ., Dordrecht, 1991. | MR 1154345 | Zbl 0735.62081

[35] M. Sart. Model selection for poisson processes with covariates. ArXiv e-prints, 2012.

[36] G. Viennet. Inequalities for absolutely regular sequences: Application to density estimation. Probab. Theory Related Fields 107 (1997) 467-492. | MR 1440142 | Zbl 0933.62029