$M$-estimation for discretely sampled diffusions

Jaya P. N. Bishwal

Teoriâ slučajnyh processov, Tome 15 (2009) no. 2, pp. 62-83.

Résumé

We study the estimation of a parameter in the nonlinear drift coefficient of a stationary ergodic diffusion process satisfying a homogeneous Itô stochastic differential equation based on discrete observations of the process, when the true model does not necessarily belong to the observer's model. Local asymptotic normality of $M$-ratio random fields are studied. Asymptotic normality of approximate $M$-estimators based on the Itô and Fisk–Stratonovich approximations of a continuous $M$-functional are obtained under a moderately increasing experimental design condition through the weak convergence of approximate $M$-ratio random fields. The derivatives of an approximate log-$M$ functional based on the Itô approximation are martingales, but the derivatives of a log-$M$ functional based on the Fisk–Stratonovich approximation are not martingales, but the average of forward and backward martingales. The averaged forward and backward martingale approximations have a faster rate of convergence than the forward martingale approximations.

Détail

Keywords: Itô stochastic differential equations, model misspecification, discrete observations, moderately increasing experimental design, approximate $M$-estimators, local asymptotic normality, robustness, weak convergence of random fields.
Mots-clés : diffusion processes

Bibliographie

[1] Y. Ait-Sahalia, “Maximum likelihood estimation of discretely sampled diffusions: a closed form approximation approach”, Econometrica, 70 (2002), 223–262

[2] M. Arato, Linear Stochastic Systems with Constant Coefficients: A Statistical Approach, Lecture Notes in Control and Information Sciences, 45, Springer, New York, 1982

[3] M. Arfi, “Nonparametric drift estimation from ergodic samples”, J. Nonparametric Statist., 5 (1995), 381–389

[4] M. Arfi, J. P. Lecoutre, “Drift estimation under strong mixing”, Pub. Inst. Stat. Stat. Univ. Paris, 38 (1994)

[5] G. Banon, “Nonparametric identification for diffusion processes”, SIAM J. Control Optim., 16 (1978), 380–395

[6] I. V. Basawa, B. L. S. Prakasa Rao, Statistical Inference for Stochastic Processes, Academic Press, New York-London, 1980

[7] B. M. Bibby, M. Sørensen, “Martingale estimation functions for discretely observed diffusion processes”, Bernoulli, 1 (1995), 17–39

[8] B. M. Bibby, M. Sørensen, “On estimation for discretely observed diffusions: A review”, Theory of Stochastic Processes, 2 (1998)

[9] J. P. N. Bishwal, “Rates of weak convergence of the approximate minimum contrast estimator for the discretely observed Ornstein–Uhlenbeck process”, Statistics and Probability Letters, 76:13 (2006), 1397–1409

[10] J. P. N. Bishwal, “A new estimating function for discretely sampled diffusions”, Random Operators and Stochastic Equations, 15 (2007), 65–88

[11] J. P. N. Bishwal, Parameter Estimation in Stochastic Differential Equations, Lecture Notes in Mathematics, 1923, Springer, Berlin, 2008

[12] J. P. N. Bishwal, A. Bose, “Rates of convergence of approximate maximum likelihood estimators in the Ornstein–Uhlenbeck process”, Comput. Math. Appl., 42 (2001), 23–38

[13] L. Broze,O. Scaillet, J. M. Zakoian, “Quasi-indirect inference for diffusion processes”, Ecometric Theory, 14 (1998), 161–186

[14] E. Clement, Inference statistique des processes de diffusion, Research Report No 9404, CREST-INSEE, Universite Paris VI, 1993

[15] E. Clement, Estimation des processes de diffusion par methodes simulees, Research Report No 9507, CREST-INSEE, Universite Paris VI, 1995

[16] E. Clement, “Estimation of diffusion processes by simulated moments methods”, Scand. J. Statist., 24:3 (1997), 353–369

[17] L. Coutin, “Estimation des coefficients d'un processes cadlag observe a des temps discrets”, Pub. Inst. Stat. Univ. Paris, 38, 1994, 87–109

[18] D. Dacunha-Castelle, D. Florens-Zmirou, “Estimation of the coefficients of a diffusion from discrete observations”, Stochastics, 19 (1986), 263–284

[19] A. Ja. Dorogovcev, “The consistency of an estimate of a parameter of a stochastic differential equation”, Theory Prob. Math. Statist., 10 (1976), 73–82

[20] D. Duffie, P. Glynn, Estimation of continuous time Markov processes sampled at random time intervals, Preprint, Graduate School of Business, Stanford University, 1997

[21] D. Florens-Zmirou, “Approximate discrete-time schemes for statistics of diffusion processes”, Statistics, 20 (1989), 547–557

[22] V. Genon-Catalot, “Maximum contrast estimation for diffusion processes arom discrete observations”, Statistics, 21 (1990), 99–116

[23] I. I. Gikhman, A. V. Skorohod, Stochastic Differential Equations, Springer, Berlin, 1972

[24] C. Gouriéroux, A. Monfort, E. Renault, “Indirect inference”, Journal of Applied Econometrics, 8 (1993), 585–618

[25] C. Gouriéroux, A. Monfort, Indirect inference for stochastic differential equatios, Discussion paper 9443, CREST-INSEE, University of Paris VI, 1994

[26] V. Harison, “Drift estimation of a certain class of diffusion processes from discrete observations”, Computers Math. Applic., 31 (1996), 121–133

[27] I. A. Ibragimov, R.Ż. Has'minskii, Statistical Estimation: Asymptotic Theory, Springer, Berlin, 1981

[28] R. A. Kasonga, “The consistency of a nonlinear least squares estimator from diffusion processes”, Stoch. Proc. Appl., 30 (1988), 263–275

[29] M. Kessler, “Estimation of an ergodic diffusion from discrete observations”, Scand. J. Statist., 24:2 (1997), 211–229

[30] M. Kessler, M. Sørensen, Estimating equations based on eigen function for a discretely observed diffusion process, Research Rep. No 332, Dept. of Theoretical Statistics, Univ. Aarhus, Denmark, 1995

[31] P. E. Kloeden, H. Platen, H. Schurz,M. Sørensen, “On the effects of discretization of estimators of drift parameters for diffusion processes”, J. Appl. Probab., 33 (1996), 1061–1076

[32] Yu. A. Kutoyants, Parameter Estimation for Stochastic Processes, Translated and edited by B.L.S. Prakasa Rao, Heldermann, Berlin, 1984

[33] Yu. A. Kutoyants, Identification of Dynamical Systems with Small Noise, Kluwer, Dordrecht, 1994

[34] Yu. A. Kutoyants, “Some problems of nonparametric estimation by observations of ergodic diffusion process”, Statist. Probab. Lett., 1996

[35] Yu. A. Kutoyants, Statistical Inference for Ergodic Diffusion Processes, Springer-Verlag, 2003

[36] V. Lanksa, “Minimum contrast estimation in diffusion processes”, J. Appl. Prob., 16 (1979), 65–75

[37] A. Le Breton, “On continuous and discrete sampling for parameter estimation in diffusion type processes”, Mathematical Programming Study, 5 (1976), 124–144

[38] I. A. McKeague, “Estimation for diffusion processes under misspecified models”, J. Appl. Probab., 21 (1984), 511–520

[39] M. N. Mishra, J. P. N. Bishwal, “Approximate maximum likelihood estimation for diffusion processes from discrete observations”, Stoch. Stoch. Reports, 52 (1995), 1–13

[40] H. T. Nguyen, T. D. Pham, “Nonparametric estimation in diffusion model by discrete sampling”, Pub. Inst. Stat. Univ. Paris, 26 (1981), 89–109

[41] A. R. Pedersen, “A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations”, Scand. J. Statist., 22 (1995), 55–71

[42] A. R.Pedersen, “Consistency and asymptotic normality of an approximate maximum likelihood estimator for discretely observed diffusion processes”, Bernoulli, 1 (1995), 257–279

[43] T. D. Pham, “Nonparametric estimation of the drift coefficient in the diffusion equation”, Math. Oper. Statist., Ser. Statistics, 12 (1981), 61–73

[44] N. Yoshida, “Asymptotic behaviour of $M$-estimator and related random field for diffusion process”, Ann. Inst. Statist. Math., 42 (1990), 221–251

[45] N. Yoshida, “Estimation for diffusion processes from discrete observations”, J. Multivariate Anal., 41 (1992), 220–242