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Estimation for misspecified ergodic diffusion processesfrom discrete observations

Published online by Cambridge University Press:  05 January 2012

Masayuki Uchida  and
Nakahiro Yoshida
Masayuki Uchida
Affiliation:
Graduate School of Engineering Science, Osaka University Toyonaka, Osaka 560-8531, Japan; [email protected] Japan Science and Technology Agency, CREST, Japan
Nakahiro Yoshida
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
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Abstract

The joint estimation of both drift and diffusion coefficient parameters is treatedunder the situation where the data are discretely observed from an ergodic diffusion processand where the statistical model may or may not include the true diffusion process.We consider the minimum contrast estimator,which is equivalent to the maximum likelihood type estimator, obtained from the contrast function based on a locally Gaussian approximation of the transition density.The asymptotic normality of the minimum contrast estimator is proved.In particular, the rate of convergence for the minimum contrast estimator of diffusion coefficient parameter in a misspecified modelis different from the one in the correctly specified parametric model.

Keywords

Diffusion processmisspecified modeldiscrete time observationsminimum contrast estimatorrate of convergence
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011 , pp. 270 - 290
Copyright
© EDP Sciences, SMAI, 2011

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References

Bibby, B.M. and Sørensen, M., Martingale estimating functions for discretely observed diffusion processes. Bernoulli 1 (1995) 17–39. CrossRef
Florens-Zmirou, D., Approximate discrete time schemes for statistics of diffusion processes. Statistics 20 (1989) 547–557. CrossRef
Genon-Catalot, V. and Jacod, J., On the estimation of the diffusion coefficient for multidimensional diffusion processes. Ann. Inst. Henri Poincaré Probab. Statist. 29 (1993) 119–151.
Gobet, E., LAN property for ergodic diffusions with discrete observations. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 711–737. CrossRef
P. Hall and C. Heyde, Martingale limit theory and its applications. Academic Press, New York (1980).
I.A. Ibragimov and R.Z. Has'minskii, Statistical estimation. Springer Verlag, New York (1981).
Kessler, M., Estimation of an ergodic diffusion from discrete observations. Scand. J. Statist. 24 (1997) 211–229.
S. Kusuoka and N. Yoshida, Malliavin calculus, geometric mixing, and expansion of diffusion functionals, Probab. Theory Relat. Fields 116 (2000) 457–484. CrossRef
Yu.A. Kutoyants, Statistical inference for ergodic diffusion processes. Springer-Verlag, London (2004).
Masuda, H., Ergodicity and exponential β-mixing bound for multidimensional diffusions with jumps. Stochastic Processes Appl.  117 (2007) 35–56. CrossRef
McKeague, I.W., Estimation for diffusion processes under misspecified models. J. Appl. Probab.  21 (1984) 511–520.
Meyn, S.P. and Tweedie, P.L., Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes. Adv. in Appl. Probab. 25 (1993) 518–548.
Pardoux, E. and Veretennikov, A.Y., On the Poisson equation and diffusion approximation 1. Ann. Prob.  29 (2001) 1061–1085.
Prakasa Rao, B.L.S., Asymptotic theory for nonlinear least squares estimator for diffusion processes. Math. Operationsforsch. Statist. Ser. Statist. 14 (1983) 195–209.
Prakasa Rao, B.L.S., Statistical inference from sampled data for stochastic processes. Contemp. Math. 80 (1988) 249–284. Amer. Math. Soc., Providence, RI. CrossRef
Uchida, M. and Yoshida, N., Information criteria in model selection for mixing processes. Statist. Infer. Stochast. Process. 4 (2001) 73–98. CrossRef
Yoshida, N., Asymptotic behavior of M-estimator and related random field for diffusion process. Ann. Inst. Statist. Math. 42 (1990) 221–251.
Yoshida, N., Estimation for diffusion processes from discrete observation. J. Multivariate Anal. 41 (1992) 220–242. CrossRef
N. Yoshida, Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations (to appear in Ann. Inst. Statist. Math.) (2005).