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van der Meulen, Frank; van Zanten, Harry

Consistent nonparametric Bayesian inference for discretely observed scalar diffusions. (English) Zbl 1259.62070

Bernoulli 19, No. 1, 44-63 (2013).
Summary: We study Bayes procedures for the problem of nonparametric drift estimation for one-dimensional, ergodic diffusion models from discrete-time, low-frequency data. We give conditions for posterior consistency and verify these conditions for concrete priors, including priors based on wavelet expansions.

Cited in 15 Documents

MSC:

62M05 Markov processes: estimation; hidden Markov models
62G05 Nonparametric estimation
62F15 Bayesian inference
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems

Keywords:

Bayesian nonparametrics; drift function; posterior consistency; posterior distribution; stochastic differential equations; wavelets
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References:

[1] Barron, A., Schervish, M.J. and Wasserman, L. (1999). The consistency of posterior distributions in nonparametric problems. Ann. Statist. 27 536-561. · Zbl 0980.62039 · doi:10.1214/aos/1018031206
[2] Beskos, A., Papaspiliopoulos, O., Roberts, G.O. and Fearnhead, P. (2006). Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 333-382. · Zbl 1100.62079 · doi:10.1111/j.1467-9868.2006.00552.x
[3] Birgé, L. (1982). Tests robustes pour des variables indépendantes et des chaînes de Markov. Ann. Sci. Univ. Clermont-Ferrand II Math. 20 70-77. · Zbl 0554.62031
[4] Borodin, A.N. and Salminen, P. (2002). Handbook of Brownian Motion-Facts and Formulae , 2nd ed. Probability and Its Applications . Basel: Birkhäuser. · Zbl 1012.60003
[5] Chib, S., Shephard, N. and Pitt, M. (2010). Likelihood based inference for diffusion driven state space models.
[6] Daubechies, I. (1992). Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics 61 . Philadelphia, PA: SIAM. · Zbl 0776.42018
[7] Diaconis, P. and Freedman, D. (1986). On the consistency of Bayes estimates. Ann. Statist. 14 1-26. · Zbl 0595.62022 · doi:10.1214/aos/1176349830
[8] Elerian, O., Chib, S. and Shephard, N. (2001). Likelihood inference for discretely observed nonlinear diffusions. Econometrica 69 959-993. · Zbl 1017.62068 · doi:10.1111/1468-0262.00226
[9] Eraker, B. (2001). MCMC analysis of diffusion models with application to finance. J. Bus. Econom. Statist. 19 177-191.
[10] Ghosal, S., Ghosh, J.K. and van der Vaart, A.W. (2000). Convergence rates of posterior distributions. Ann. Statist. 28 500-531. · Zbl 1105.62315 · doi:10.1214/aos/1016218228
[11] Ghosal, S. and Tang, Y. (2006). Bayesian consistency for Markov processes. Sankhyā 68 227-239. · Zbl 1193.62035
[12] Ghosal, S. and van der Vaart, A. (2007). Convergence rates of posterior distributions for non-i.i.d. observations. Ann. Statist. 35 192-223. · Zbl 1114.62060 · doi:10.1214/009053606000001172
[13] Golightly, A. and Wilkinson, D.J. (2008). Bayesian inference for nonlinear multivariate diffusion models observed with error. Comput. Statist. Data Anal. 52 1674-1693. · Zbl 1452.62603
[14] Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets , Approximation , and Statistical Applications. Lecture Notes in Statistics 129 . New York: Springer. · Zbl 0899.62002
[15] Hernández, E. and Weiss, G. (1996). A First Course on Wavelets. Studies in Advanced Mathematics . Boca Raton, FL: CRC Press. With a foreword by Yves Meyer. · Zbl 0885.42018
[16] Itô, K. and McKean, H.P. Jr. (1965). Diffusion Processes and Their Sample Paths . Berlin: Springer. · Zbl 0127.09503
[17] Jensen, B. and Poulsen, R. (2002). Transition densities of diffusion processes: Numerical comparison of approximation techniques. Journal of Derivatives 9 1-15.
[18] Kallenberg, O. (1997). Foundations of Modern Probability. Probability and Its Applications ( New York ). New York: Springer. · Zbl 0892.60001
[19] Karatzas, I. and Shreve, S.E. (1991). Brownian Motion and Stochastic Calculus , 2nd ed. Graduate Texts in Mathematics 113 . New York: Springer. · Zbl 0734.60060
[20] Roberts, G.O. and Stramer, O. (2001). On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm. Biometrika 88 603-621. · Zbl 0985.62066 · doi:10.1093/biomet/88.3.603
[21] Rogers, L.C.G. and Williams, D. (1987). Diffusions , Markov Processes , and Martingales. Vol. 2, Itô Calculus. Wiley Series in Probability and Mathematical Statistics : Probability and Mathematical Statistics . New York: Wiley. · Zbl 0977.60005
[22] Schwartz, L. (1965). On Bayes procedures. Z. Wahrsch. Verw. Gebiete 4 10-26. · Zbl 0158.17606 · doi:10.1007/BF00535479
[23] Shen, X. and Wasserman, L. (2001). Rates of convergence of posterior distributions. Ann. Statist. 29 687-714. · Zbl 1041.62022 · doi:10.1214/aos/1009210686
[24] Tang, Y. and Ghosal, S. (2007). Posterior consistency of Dirichlet mixtures for estimating a transition density. J. Statist. Plann. Inference 137 1711-1726. · Zbl 1118.62089 · doi:10.1016/j.jspi.2006.03.007
[25] Walker, S. (2004). New approaches to Bayesian consistency. Ann. Statist. 32 2028-2043. · Zbl 1056.62040 · doi:10.1214/009053604000000409
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