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Inhomogeneous and anisotropic conditional density estimation from dependent data. (English) Zbl 1271.62060
Summary: The problem of estimating a conditional density is considered. Given a collection of partitions, we propose a procedure that selects from the data the best partition among that collection and then provides the best piecewise polynomial estimator built on that partition. The observations are not supposed to be independent but only \(\beta\)-mixing; in particular, our study includes the estimation of the transition density of a Markov chain. For a well-chosen collection of possibly irregular partitions, we obtain oracle-type inequalities and adaptivity results in the minimax sense over a wide range of possibly anisotropic and inhomogeneous Besov classes. We end with a short simulation study.

MSC:
62G05 Nonparametric estimation
62H12 Estimation in multivariate analysis
62M05 Markov processes: estimation; hidden Markov models
62M09 Non-Markovian processes: estimation
Software:
CAPUSHE
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References:
[1] N. Akakpo., Estimation adaptative par sélection de partitions en rectangles dyadiques . PhD thesis, Université Paris-Sud 11, 2009.
[2] N. Akakpo. Adaptation to anisotropy and inhomogeneity via dyadic piecewise polynomial selection., 1102.3108, 2011. · Zbl 1308.62070
[3] P. Ango Nze. Critères d’ergodicité géométrique ou arithmétique de modèles linéaires perturbés à représentation markovienne., C. R. Acad. Sci. Paris Sér. I Math. , 326(3):371-376, 1998. · Zbl 0918.60052
[4] A. Barron, L. Birgé, and P. Massart. Risk bounds for model selection via penalization., Probab. Theory Related Fields , 113(3): 301-413, 1999. · Zbl 0946.62036
[5] E. Brunel, F. Comte, and C. Lacour. Adaptive Estimation of the Conditional Density in Presence of Censoring., Sankhyā , 69(4): 734-763, 2007. · Zbl 1193.62055
[6] L. Birgé. Approximation dans les espaces metriques et theorie de l’estimation., Probability Theory and Related Fields , 65(2):181-237, 1983. · Zbl 0506.62026
[7] L. Birgé and P. Massart. From model selection to adaptive estimation. In, Festschrift for Lucien Le Cam , pages 55-87. Springer, New York, 1997. · Zbl 0920.62042
[8] J.P. Baudry, C. Maugis, and B. Michel. Slope heuristics: overview and implementation., Statistics and Computing , 22(2):455-470, 2012. · Zbl 1322.62007
[9] D. Bosq., Nonparametric statistics for stochastic processes , volume 110 of Lecture Notes in Statistics . Springer-Verlag, New York, second edition, 1998. Estimation and prediction. · Zbl 0902.62099
[10] R. C. Bradley. Basic properties of strong mixing conditions. A survey and some open questions., Probab. Surv. , 2:107-144 (electronic), 2005. Update of, and a supplement to, the 1986 original. · Zbl 1189.60077
[11] R. C. Bradley., Introduction to strong mixing conditions. Vol. 1 . Kendrick Press, Heber City, UT, 2007. · Zbl 1133.60001
[12] G. Blanchard, C. Schäfer, Y. Rozenholc, and K.R. Müller. Optimal dyadic decision trees., Machine Learning , 66(2): 209-241, 2007.
[13] S. Clémençon., Méthodes d’ondelettes pour la statistique non paramétrique des chaînes de Markov . PhD thesis, Doctoral thesis, Université Paris VII, 2000.
[14] S. J. M. Clémençon. Adaptive estimation of the transition density of a regular Markov chain., Math. Methods Statist. , 9(4):323-357, 2000. · Zbl 1008.62076
[15] F. Comte and F. Merlevède. Adaptive estimation of the stationary density of discrete and continuous time mixing processes., ESAIM Probab. Statist. , 6:211-238 (electronic), 2002.
[16] P. Doukhan and M. Ghindès. Estimation de la transition de probabilité d’une chaîne de Markov Doëblin-récurrente. Étude du cas du processus autorégressif général d’ordre 1., Stochastic Process. Appl. , 15(3):271-293, 1983. · Zbl 0515.62037
[17] J. G. De Gooijer and D. Zerom. On conditional density estimation., Statist. Neerlandica , 57(2):159-176, 2003. · Zbl 1090.62526
[18] D. L. Donoho. CART and best-ortho-basis: a connection., Ann. Statist. , 25(5) :1870-1911, 1997. · Zbl 0942.62044
[19] P. Doukhan., Mixing , volume 85 of Lecture Notes in Statistics . Springer-Verlag, New York, 1994. Properties and examples.
[20] J. Dedecker and C. Prieur. New dependence coefficients. Examples and applications to statistics., Probab. Theory Related Fields , 132(2):203-236, 2005. · Zbl 1061.62058
[21] P. Doukhan and A. B. Tsybakov. Nonparametric recurrent estimation in nonlinear ARX models., Problemy Peredachi Informatsii , 29(4):24-34, 1993. · Zbl 0804.62041
[22] S. Efromovich. Conditional density estimation in a regression setting., Ann. Statist. , 35(6) :2504-2535, 2007. · Zbl 1129.62025
[23] S. Efromovich. Oracle inequality for conditional density estimation and an actuarial example., Annals of the Institute of Statistical Mathematics , 62(2):249-275, 2010. · Zbl 1440.62118
[24] J. Engel. A simple wavelet approach to nonparametric regression from recursive partitioning schemes., J. Multivariate Anal. , 49(2): 242-254, 1994. · Zbl 0795.62034
[25] J. Engel. The multiresolution histogram., Metrika , 46(1):41-57, 1997. · Zbl 0872.62041
[26] O. P. Faugeras. A quantile-copula approach to conditional density estimation., J. Multivariate Anal. , 100(9) :2083-2099, 2009. · Zbl 1170.62030
[27] J. Fan and T. H. Yim. A crossvalidation method for estimating conditional densities., Biometrika , 91(4):819-834, 2004. · Zbl 1078.62032
[28] L. Györfi and M. Kohler. Nonparametric estimation of conditional distributions., IEEE Trans. Inform. Theory , 53(5) :1872-1879, 2007. · Zbl 1316.62047
[29] I. Gannaz and O. Wintenberger. Adaptive density estimation under weak dependence., ESAIM Probab. Statist. , 14:151-172, 2010. · Zbl 1209.62056
[30] P. Hall, G. Kerkyacharian, and D. Picard. Block threshold rules for curve estimation using kernel and wavelet methods., Ann. Statist. , 26(3):922-942, 1998. · Zbl 0929.62040
[31] G. L. Jones. On the Markov chain central limit theorem., Probab. Surv. , 1:299-320 (electronic), 2004. · Zbl 1189.60129
[32] J. Klemelä. Multivariate histograms with data-dependent partitions., Statist. Sinica , 19(1):159-176, 2009. · Zbl 1153.62047
[33] C. Lacour. Adaptive estimation of the transition density of a Markov chain., Ann. Inst. H. Poincaré Probab. Statist. , 43(5):571-597, 2007. · Zbl 1125.62087
[34] P. Massart., Concentration inequalities and model selection , volume 1896 of Lecture Notes in Mathematics . Springer, Berlin, 2007. Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6-23, 2003, With a foreword by Jean Picard. · Zbl 1170.60006
[35] A. Mokkadem. Propriétés de mélange des processus autorégressifs polynomiaux., Ann. Inst. H. Poincaré Probab. Statist. , 26(2):219-260, 1990. · Zbl 0706.60040
[36] S. P. Meyn and R. L. Tweedie., Markov chains and stochastic stability . Communications and Control Engineering Series. Springer-Verlag London Ltd., London, 1993. · Zbl 0925.60001
[37] G. G. Roussas. Nonparametric estimation in markov processes., Annals of the Institute of Statistical Mathematics , 21(1):73-87, 1969. · Zbl 0181.45804
[38] G. O. Roberts and J. S. Rosenthal. Geometric ergodicity and hybrid Markov chains., Electron. Comm. Probab. , 2:no. 2, 13-25 (electronic), 1997. · Zbl 0890.60061
[39] G. O. Roberts and J. S. Rosenthal. Variance bounding Markov chains., Ann. Appl. Probab. , 18(3) :1201-1214, 2008. · Zbl 1142.60047
[40] Richard P. Stanley., Enumerative combinatorics. Vol. 2 , volume 62 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin.
[41] H. Triebel., Theory of function spaces. III , volume 100 of Monographs in Mathematics . Birkhäuser Verlag, Basel, 2006. · Zbl 1104.46001
[42] G. Viennet. Inequalities for absolutely regular sequences: application to density estimation., Probab. Theory Related Fields , 107(4):467-492, 1997. · Zbl 0933.62029
[43] R. M. Willett and Robert D. Nowak. Multiscale Poisson intensity and density estimation., IEEE Trans. Inform. Theory , 53(9): 3171-3187, 2007. · Zbl 1325.94036
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