Estimation for misspecified ergodic diffusion processes from discrete observations

Masayuki Uchida; Nakahiro Yoshida

ESAIM: Probability and Statistics (2011)

  • Volume: 15, page 270-290
  • ISSN: 1292-8100

Abstract

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The joint estimation of both drift and diffusion coefficient parameters is treated under the situation where the data are discretely observed from an ergodic diffusion process and 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 model is different from the one in the correctly specified parametric model.

How to cite

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Uchida, Masayuki, and Yoshida, Nakahiro. "Estimation for misspecified ergodic diffusion processes from discrete observations." ESAIM: Probability and Statistics 15 (2011): 270-290. <http://eudml.org/doc/277159>.

@article{Uchida2011,
abstract = {The joint estimation of both drift and diffusion coefficient parameters is treated under the situation where the data are discretely observed from an ergodic diffusion process and 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 model is different from the one in the correctly specified parametric model.},
author = {Uchida, Masayuki, Yoshida, Nakahiro},
journal = {ESAIM: Probability and Statistics},
keywords = {diffusion process; misspecified model; discrete time observations; minimum contrast estimator; rate of convergence},
language = {eng},
pages = {270-290},
publisher = {EDP-Sciences},
title = {Estimation for misspecified ergodic diffusion processes from discrete observations},
url = {http://eudml.org/doc/277159},
volume = {15},
year = {2011},
}

TY - JOUR
AU - Uchida, Masayuki
AU - Yoshida, Nakahiro
TI - Estimation for misspecified ergodic diffusion processes from discrete observations
JO - ESAIM: Probability and Statistics
PY - 2011
PB - EDP-Sciences
VL - 15
SP - 270
EP - 290
AB - The joint estimation of both drift and diffusion coefficient parameters is treated under the situation where the data are discretely observed from an ergodic diffusion process and 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 model is different from the one in the correctly specified parametric model.
LA - eng
KW - diffusion process; misspecified model; discrete time observations; minimum contrast estimator; rate of convergence
UR - http://eudml.org/doc/277159
ER -

References

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  1. [1] B.M. Bibby and M. Sørensen, Martingale estimating functions for discretely observed diffusion processes. Bernoulli1 (1995) 17–39. Zbl0830.62075MR1354454
  2. [2] D. Florens-Zmirou, Approximate discrete time schemes for statistics of diffusion processes. Statistics20 (1989) 547–557. Zbl0704.62072MR1047222
  3. [3] V. Genon-Catalot and J. Jacod, On the estimation of the diffusion coefficient for multidimensional diffusion processes. Ann. Inst. Henri Poincaré Probab. Statist. 29 (1993) 119–151. Zbl0770.62070MR1204521
  4. [4] E. Gobet, LAN property for ergodic diffusions with discrete observations. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 711–737. Zbl1018.60076MR1931584
  5. [5] P. Hall and C. Heyde, Martingale limit theory and its applications. Academic Press, New York (1980). Zbl0462.60045MR624435
  6. [6] I.A. Ibragimov and R.Z. Has'minskii, Statistical estimation. Springer Verlag, New York (1981). MR620321
  7. [7] M. Kessler, Estimation of an ergodic diffusion from discrete observations. Scand. J. Statist. 24 (1997) 211–229. Zbl0879.60058MR1455868
  8. [8] S. Kusuoka and N. Yoshida, Malliavin calculus, geometric mixing, and expansion of diffusion functionals, Probab. Theory Relat. Fields 116 (2000) 457–484. Zbl0970.60061MR1757596
  9. [9] Yu.A. Kutoyants, Statistical inference for ergodic diffusion processes. Springer-Verlag, London (2004). Zbl1038.62073MR2144185
  10. [10] H. Masuda, Ergodicity and exponential β-mixing bound for multidimensional diffusions with jumps. Stochastic Processes Appl.  117 (2007) 35–56. Zbl1118.60070MR2287102
  11. [11] I.W. McKeague, Estimation for diffusion processes under misspecified models. J. Appl. Probab.  21 (1984) 511–520. Zbl0555.62067MR752016
  12. [12] S.P. Meyn and P.L. Tweedie, Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes. Adv. in Appl. Probab. 25 (1993) 518–548. Zbl0781.60053MR1234295
  13. [13] E. Pardoux and A.Y. Veretennikov, On the Poisson equation and diffusion approximation 1. Ann. Prob.  29 (2001) 1061–1085. Zbl1029.60053MR1872736
  14. [14] B.L.S. Prakasa Rao, Asymptotic theory for nonlinear least squares estimator for diffusion processes. Math. Operationsforsch. Statist. Ser. Statist. 14 (1983) 195–209. Zbl0532.62060MR704787
  15. [15] B.L.S. Prakasa Rao, Statistical inference from sampled data for stochastic processes. Contemp. Math. 80 (1988) 249–284. Amer. Math. Soc., Providence, RI. Zbl0687.62069MR999016
  16. [16] M. Uchida and N. Yoshida, Information criteria in model selection for mixing processes. Statist. Infer. Stochast. Process. 4 (2001) 73–98. Zbl1092.62595MR1850590
  17. [17] N. Yoshida, Asymptotic behavior of M-estimator and related random field for diffusion process. Ann. Inst. Statist. Math. 42 (1990) 221–251. Zbl0723.62048MR1064786
  18. [18] N. Yoshida, Estimation for diffusion processes from discrete observation. J. Multivariate Anal. 41 (1992) 220–242. Zbl0811.62083MR1172898
  19. [19] N. Yoshida, Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations (to appear in Ann. Inst. Statist. Math.) (2005). MR2786943

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