Abstract
Methods of robust estimation in diffusion processes are given by means of M-estimation. It is shown that the asymptotic variance of an M-estimator is obtained by applying a certain integral operator to the influence function and integrating its square. Under the condition of boundedness of the influence function, the existence of an optimal robust M-estimator is shown and an approximately optimal practical method is given. Moreover, as another criterion of robustness we consider a norm of integral type and show that the corresponding optimal robust M-estimator is obtained by solving a boundary value problem of a second order differential equation. Finally, as an illustrative example the Ornstein-Uhlenbeck process is discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bustos, O. (1982). General M-estimates for contaminated pth-order autoregressive processes: Consistency and asymptotic normality, Z. Wahrsch. Verw. Gebiete, 59, 491–504.
Bustos, O. and Yohai, V. J. (1986). Robust estimates for ARMA models, J. Amer. Statist. Assoc., 81, 155–168.
Denby, L. and Martin, R. D. (1979). Robust estimation of the first order autoregressive parameter, J. Amer. Statist. Assoc., 74, 140–146.
Feigin, P. D. (1985). Stable convergence of semimartingales, Stochastic Process. Appl., 19, 125–134.
Gihman, I. I. and Skorohod, A. V. (1972). Stochastic Differential Equations, Springer, Berlin-New York.
Hampel, F. R. (1974). The influence curve and its role in robust estimation, J. Amer. Statist. Assoc., 69, 383–393.
Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A. (1986). Robust Statistics, Wiley, New York.
Huber, P. J. (1981). Robust Statistics, Wiley, New York.
Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam-New York.
Itô, K. and McKean, H. P.Jr. (1965). Diffusion Processes and Their Sample Paths, Springer, Berlin-New York.
Kleiner, B., Martin, R. D. and Thomson, D. J. (1979). Robust estimation of power spectra, J. Roy. Statist. Soc. Ser. B, 41, 313–351.
Krylov, N. V. (1980). Controlled Diffusion Processes, Springer, Berlin-New York.
Künsch, H. (1984). Infinitesimal robustness for autoregressive processes, Ann. Statist., 12, 843–863.
Kutoyants, Yu. A. (1977). Estimation of the trend parameter of a diffusion in the smooth case, Theory Probab. Appl., 22, 299–405.
Kutoyants, Yu. A. (1978). Estimation of a parameter of a diffusion process, Theory Probab. Appl., 23, 641–649.
Kutoyants, Yu. A. (1984). Parameter Estimation for Stochastic Processes, (trans. and ed. B. L. S., Prakasa Rao), Heldermann Verlag, Berlin.
Lánska, V. (1979). Minimum contrast estimation in diffusion processes. J. Appl. Probab., 16, 65–75.
Lepingle, D. (1978). Sur le Comportement Asymptotique des Martingales Locales, Lecture Notes in Mathematics, 649, 148–161, Springer, Berlin-New York.
Liptser, R. Sh. and Shiryayev, A. N. (1977). Statistics of Random Processes, Vol. I, Springer, Berlin-New York.
Mandl, P. (1968) Analytic Treatment of One-Dimensional Markov Processes, Springer, Prague.
Martin, R. D. and Yohai, V. J. (1985). Robustness in time series and estimating ARMA models, Handbook of Statistics, 5, 119–155.
Martin, R. D. and Yohai, V. J. (1986). Influence functionals for time series, Ann. Statist., 14, 781–818.
McKeague, I. W. (1984). Estimation for diffusion processes under misspecified models, J. Appl. Probab., 21, 511–520.
Powell, M. J. D. (ed.) (1982). Nonlinear Optimization 1981, Academic Press, New York.
Rao, B. L. S. P. and Rubin, H. (1981). Asymptotic theory of estimation in non-linear stochastic differential equations, Sankhyā Ser. A, 43, 170–189.
Author information
Authors and Affiliations
About this article
Cite this article
Yoshida, N. Robust M-estimators in diffusion processes. Ann Inst Stat Math 40, 799–820 (1988). https://doi.org/10.1007/BF00049433
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF00049433