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Nonparametric Bayesian inference for ergodic diffusions. (English) Zbl 1183.62144

Summary: The problem of nonparametric drift estimation for ergodic diffusions is studied from a Bayesian perspective. In particular, Gaussian process priors are exhibited that yield optimal contraction rates if the drift function belongs to a smoothness class.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62F15 Bayesian inference
62G05 Nonparametric estimation
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References:

[1] Banon, G., Nonparametric identification for diffusion processes, SIAM J. Control Optim., 16, 3, 380-395 (1978) · Zbl 0404.93045
[2] Belitser, E.; Ghosal, S., Adaptive Bayesian inference on the mean of an infinite-dimensional normal distribution, Ann. Statist., 31, 2, 536-559 (2003) · Zbl 1039.62039
[3] Castillo, I., 2008. Lower bounds for posterior rates with Gaussian process priors. Preprint.; Castillo, I., 2008. Lower bounds for posterior rates with Gaussian process priors. Preprint. · Zbl 1320.62067
[4] Dalalyan, A., Sharp adaptive estimation of the drift function for ergodic diffusions, Ann. Statist., 33, 6, 2507-2528 (2005) · Zbl 1084.62079
[5] Diaconis, P.; Freedman, D., On the consistency of Bayes estimates, Ann. Statist., 14, 1, 1-67 (1986) · Zbl 0595.62022
[6] Galtchouk, L.; Pergamenshchikov, S., Sequential nonparametric adaptive estimation of the drift coefficient in diffusion processes, Math. Methods Statist., 10, 3, 316-330 (2001) · Zbl 1005.62070
[7] Ghosal, S.; Ghosh, J. K.; Van der Vaart, A. W., Convergence rates of posterior distributions, Ann. Statist., 28, 2, 500-531 (2000) · Zbl 1105.62315
[8] Ghosal, S.; Lember, J.; Van der Vaart, A. W., Nonparametric Bayesian model selection and averaging, Electron. J. Stat., 2, 63-89 (2008) · Zbl 1135.62028
[9] Ghosal, S.; Van der Vaart, A. W., Convergence rates of posterior distributions for non-i.i.d. observations, Ann. Statist., 35, 1, 192-223 (2007) · Zbl 1114.62060
[10] Huang, T.-M., Convergence rates for posterior distributions and adaptive estimation, Ann. Statist., 32, 4, 1556-1593 (2004) · Zbl 1095.62055
[11] Itô, K.; McKean, H. P., Diffusion Processes and their Sample Paths (1965), Springer: Springer Berlin · Zbl 0127.09503
[12] Karatzas, I.; Shreve, S. E., Brownian Motion and Stochastic Calculus (1991), Springer: Springer New York · Zbl 0734.60060
[13] Kleijn, B. J.K.; Van der Vaart, A. W., Misspecification in infinite-dimensional Bayesian statistics, Ann. Statist., 34, 2, 837-877 (2006) · Zbl 1095.62031
[14] Kutoyants, Y. A., Statistical inference for ergodic diffusion processes, (Springer Series in Statistics (2004), Springer: Springer London) · Zbl 1288.62028
[15] Liptser, R. S.; Shiryayev, A. N., Statistics of Random Processes I (1977), Springer: Springer New York · Zbl 0364.60004
[16] Spokoiny, V. G., Adaptive drift estimation for nonparametric diffusion model, Ann. Statist., 28, 3, 815-836 (2000) · Zbl 1105.62330
[17] Tuan, P. D., Nonparametric estimation of the drift coefficient in the diffusion equation, Math. Operationsforsch. Statist. Ser. Statist., 12, 1, 61-73 (1981) · Zbl 0485.62089
[18] Van der Meulen, F. H.; Van der Vaart, A. W.; Van Zanten, J. H., Convergence rates of posterior distributions for Brownian semimartingale models, Bernoulli, 12, 5, 863-888 (2006) · Zbl 1142.62057
[19] Van der Vaart, A. W.; Van Zanten, J. H., Rates of contraction of posterior distributions based on Gaussian process priors, Ann. Statist., 36, 3, 1435-1463 (2008) · Zbl 1141.60018
[20] Van der Vaart, A. W.; Van Zanten, J. H., Reproducing kernel Hilbert spaces of Gaussian priors, IMS Collections, 3, 200-222 (2008) · Zbl 1141.60018
[21] Van der Vaart, A.W., Van Zanten, J.H., 2009. Adapative Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth. Ann. Statist., to appear.; Van der Vaart, A.W., Van Zanten, J.H., 2009. Adapative Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth. Ann. Statist., to appear. · Zbl 1173.62021
[22] Van Zanten, J. H., Rates of convergence and asymptotic normality of kernel estimators for ergodic diffusion processes, J. Nonparametr. Statist., 13, 6, 833-850 (2001) · Zbl 0999.62030
[23] Van Zanten, J. H., On uniform laws of large numbers for ergodic diffusions and consistency of estimators, Stat. Inference Stoch. Process., 6, 2, 199-213 (2003) · Zbl 1036.60025
[24] Wahba, G., Improper priors, spline smoothing and the problem of guarding against model errors in regression, J. Roy. Statist. Soc. Ser. B, 40, 3, 364-372 (1978) · Zbl 0407.62048
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