Abstract
The M-estimate which maximizes a positive stochastic process Q is treated for multidimensional diffusion models. The convergence in distribution of the process of ratio of Q's after normalizing is proved. The asymptotic behavior of M-estimates is stated. We present the asymptotic variance in general cases and in estimation by misspecified models.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aldous, D. J. and Eagleson, G. K. (1978). On mixing and stability of limit theorems, Ann. Probab., 6, 325–331.
Arnold, I. and Kliemann, W. (1987). On unique ergodicity for degenerate diffusions, Stochastics, 21, 41–61.
Basawa, I. V. and Scott, D. J. (1983). Asymptotic optimal inference for non-ergodic models, Lecture Notes in Statist., 17, Springer, New York.
Bustos, O. (1982). General M-estimates for contaminated pth-order autoregressive processes: consistency and asymptotic normality, Z. Wahrsch. Verw. Gebiete, 59, 491–504.
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, New York.
Ibragimov, I. A. and Has'minskii, R. Z. (1972). Asymptotic behavior of statistical estimators in the smooth case, Theory Probab. Appl., 17, 443–460.
Ibragimov, I. A. and Has'minskii, R. Z. (1973). Asymptotic behavior of some statistical estimators II, Limit theorem for the a posteriori density and Bayes estimators, Theory Probab. Appl., 18, 76–91.
Ibragimov, I. A. and Has'minskii, R. Z. (1981). Statistical Estimation, Springer, New York.
Inagaki, N. and Ogata, Y. (1975). The weak convergence of likelihood ratio random fields and applications, Ann. Inst. Statist. Math., 27, 391–419.
Jeganathan, P. (1982a). On the asymptotic theory of statistical estimation when the limit of the log-likelihood ratios is mixed normal, Sankhyā Ser. A, 44, 173–212.
Jeganathan, P. (1982b). On the convergence of moments of statistical estimators, Sankhyā Ser. A, 44, 213–232.
Keller, G., Kersting, G. and Rösler, U. (1984). On the asymptotic behaviour of solutions of stochastic differential equations. Z. Wahrsch. Verw. Gebiete, 68, 163–189.
Kutoyants, Yu. A. (1977). Estimation of the drift coefficient parameter of a diffusion in the smooth case, Theory Probab. Appl., 22, 399–406.
Kutoyants, Yu. A. (1978). Estimation of a parameter of a diffusion processes, 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.
Lanska, V. (1979). Minimum contrast estimation in diffusion processes, J. Appl. Probab., 16, 65–75.
LeCam, L. (1960). Locally asymptotic normal family of distributions, Univ. Calif. Publ. Statist., 3, 27–98.
Liptser, R. S. and Shiryayev, A. N. (1977). Statistics of Random Processes, Springer, New York.
McShane, E. J. (1972). Stochastic differential equations and models of random processes, Proc. Sixth Berkeley Symp. on Math. Statist. Prob., Vol. 3, 263–294, Univ. of California Press, Berkeley.
McShane, E. J. (1974). Stochastic Calculus and Stochastic Models, Academic Press, New York.
Novikov, A. A. (1971). On moment inequalities for stochastic integrals, Theory Probab. Appl., 16, 538–541.
Ogata, Y. and Inagaki, N. (1977). The weak convergence of the likelihood ratio random fields for Markov observations, Ann. Inst. Statist. Math. 29, 165–187.
Strasser, H. (1985). Mathematical Theory of Statistics, Walter de Gruyter, Berlin.
Wong, E. and Zakai, M. (1965). On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist., 36, 1560–1564.
Yoshida, N. (1987). Asymptotic behavior of likelihood ratio in stochastic process (in Japanese), Master Thesis, Osaka University.
Yoshida, N. (1988a). Robust M-estimators in diffusion processes, Ann. Inst. Statist. Math., 40, 799–820.
Yoshida, N. (1988b). Asymptotic behavior of M-estimates in diffusion processes, Research Reports on Statistics, 22, Osaka Univ.
Author information
Authors and Affiliations
About this article
Cite this article
Yoshida, N. Asymptotic behavior of M-estimator and related random field for diffusion process. Ann Inst Stat Math 42, 221–251 (1990). https://doi.org/10.1007/BF00050834
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00050834