The Annals of Statistics

Sharp adaptive estimation of the drift function for ergodic diffusions

Arnak Dalalyan

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Abstract

The global estimation problem of the drift function is considered for a large class of ergodic diffusion processes. The unknown drift S(⋅) is supposed to belong to a nonparametric class of smooth functions of order k≥1, but the value of k is not known to the statistician. A fully data-driven procedure of estimating the drift function is proposed, using the estimated risk minimization method. The sharp adaptivity of this procedure is proven up to an optimal constant, when the quality of the estimation is measured by the integrated squared error weighted by the square of the invariant density.

Article information

Source
Ann. Statist., Volume 33, Number 6 (2005), 2507-2528.

Dates
First available in Project Euclid: 17 February 2006

Permanent link to this document
https://projecteuclid.org/euclid.aos/1140191664

Digital Object Identifier
doi:10.1214/009053605000000615

Mathematical Reviews number (MathSciNet)
MR2253093

Zentralblatt MATH identifier
1084.62079

Subjects
Primary: 62M05: Markov processes: estimation 62G07: Density estimation 62G20: Asymptotic properties

Keywords
Ergodic diffusion invariant density minimax drift estimation Pinsker’s constant sharp adaptivity

Citation

Dalalyan, Arnak. Sharp adaptive estimation of the drift function for ergodic diffusions. Ann. Statist. 33 (2005), no. 6, 2507--2528. doi:10.1214/009053605000000615. https://projecteuclid.org/euclid.aos/1140191664


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References

  • Aït-Sahalia, Y. (2002). Maximum-likelihood estimation of discretely-sampled diffusions: A closed-form approximation approach. Econometrica 70 223–262.
  • Aït-Sahalia, Y. and Mykland, P. (2004). Estimators of diffusions with randomly spaced discrete observations: A general theory. Ann. Statist. 32 2186–2222.
  • Banon, G. (1978). Nonparametric identification for diffusion processes. SIAM J. Control Optim. 16 380–395.
  • Cavalier, L., Golubev, G. K., Picard, D. and Tsybakov, A. B. (2002). Oracle inequalities for inverse problems. Ann. Statist. 30 843–874.
  • Dalalyan, A. S. (2002). Sharp adaptive estimation of the trend coefficient for ergodic diffusion. Prépublication no. 02-1, Université du Maine. Available at www.univ-lemans.fr/sciences/statist/liens/publications.html.
  • Dalalyan, A. S. and Kutoyants, Yu. A. (2002). Asymptotically efficient trend coefficient estimation for ergodic diffusion. Math. Methods Statist. 11 402–427.
  • Delattre, S. and Hoffmann, M. (2002). Asymptotic equivalence for a null recurrent diffusion. Bernoulli 8 139–174.
  • Delattre, S., Hoffmann, M. and Kessler, M. (2002). Dynamics adaptive estimation of a scalar diffusion. Prépublication PMA-762, Univ. Paris 6. Available at www.proba.jussieu.fr/mathdoc/preprints/.
  • Efromovich, S. Yu. (1985). Non-parametric estimation of a density with unknown smoothness. Theory Probab. Appl. 30 557–568.
  • Efromovich, S. Yu. (1999). Nonparametric Curve Estimation. Springer, New York.
  • Efromovich, S. Yu. and Pinsker, M. S. (1984). A self-training algorithm for nonparametric filtering. Autom. Remote Control 1984(11) 58–65.
  • Fan, J. (1991). On the estimation of quadratic functionals. Ann. Statist. 19 1273–1294.
  • Fan, J. (1993). Local linear regression smoothers and their minimax efficiencies. Ann. Statist. 21 196–216.
  • Fan, J. (2005). A selective overview of nonparametric methods in financial econometrics (with discussion). Statist. Sci. 20 317–357.
  • Fan, J. and Zhang, C. (2003). A re-examination of diffusion estimators with applications to financial model validation. J. Amer. Statist. Assoc. 98 118–134.
  • Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ.
  • Galtchouk, L. and Pergamenshchikov, S. (2001). Sequential nonparametric adaptive estimation of the drift coefficient in diffusion processes. Math. Methods Statist. 10 316–330.
  • Gihman, I. I. and Skorohod, A. V. (1972). Stochastic Differential Equations. Springer, New York.
  • Golubev, G. K. (1987). Adaptive asymptotically minimax estimates of smooth signals. Problems Inform. Transmission 23(1) 57–67.
  • Golubev, G. K. (1991). Local asymptotic normality in problems of nonparametric estimation of functions and lower bounds for quadratic risks. Theory Probab. Appl. 36 152–157.
  • Golubev, G. K. (1992). Non-parametric estimation of smooth densities of a distribution in $L^2$. Problems Inform. Transmission 28(1) 44–54.
  • Golubev, G. K. and Nussbaum, M. (1992). Adaptive spline estimates in a nonparametric regression model. Theory Probab. Appl. 37 521–529.
  • Hoffmann, M. (1999). Adaptive estimation in diffusion processes. Stochastic Process. Appl. 79 135–163.
  • Jacod, J. (2001). Inference for stochastic processes. Prépublication PMA-683, Univ. Paris 6. Available at www.proba.jussieu.fr/mathdoc/preprints/.
  • Jiang, G. J. and Knight, J. (1997). A nonparametric approach to the estimation of diffusion processes–-with an application to a short-term interest rate model. Econometric Theory 13 615–645.
  • Kutoyants, Yu. A. (1998). Efficient density estimation for ergodic diffusion processes. Stat. Inference Stoch. Process. 1 131–155.
  • Kutoyants, Yu. A. (2004). Statistical Inference for Ergodic Diffusion Processes. Springer, New York.
  • Lepskii, O. V. (1991). Asymptotically minimax adaptive estimation. I. Upper bounds. Theory Probab. Appl. 36 682–697.
  • Milstein, G. and Nussbaum, M. (1998). Diffusion approximation for nonparametric autoregression. Probab. Theory Related Fields 112 535–543.
  • Nelson, D. B. (1990). ARCH models as diffusion approximations. J. Econometrics 45 7–38.
  • Pham, T. D. (1981). Nonparametric estimation of the drift coefficient in the diffusion equation. Math. Operationsforsch. Statist. Ser. Statist. 12 61–73.
  • Pinsker, M. S. (1980). Optimal filtration of square-integrable signals in Gaussian noise. Problems Inform. Transmission 16(2) 52–68.
  • Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer, Berlin.
  • Spokoiny, V. G. (2000). Adaptive drift estimation for nonparametric diffusion model. Ann. Statist. 28 815–836.
  • van Zanten, H. (2001). Rates of convergence and asymptotic normality of kernel estimators for ergodic diffusion processes. J. Nonparametr. Statist. 13 833–850.