Abstract
The estimate of the probability of the large deviation or the statistical random field is the key to ensure the convergence of moments of the associated estimator, and it also plays an essential role to prove mathematical validity of the asymptotic expansion of the estimator. For non-linear stochastic processes, it involves technical difficulties to show a standard exponential type estimate; besides, it is not necessary for these purposes. In this paper, we propose a polynomial-type large deviation inequality which is easily verified by the L p-boundedness of certain functionals; usually they are simple additive functionals. We treat a statistical random field with multi-grades and discuss M and Bayesian type estimators. As an application, we show the behavior of those estimators, including convergence of moments, for the statistical random field in the quasi-likelihood analysis of the stochastic differential equation that is possibly multi-dimensional and non-linear. The results are new even for stochastic differential equations, while they obviously apply to other various statistical models.
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References
Adams, R. A. (1975). Sobolev spaces. In Pure and applied mathematics (Vol. 65). London, New York: Academic Press.
Arnold L., Imkeller P. (1996) Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory. Stochastic Processes and their Applications 62: 19–54
Basawa I.V., Prakasa Rao B.L.S. (1980) Statistical inference for stochastic processes. Academic Press, London, New York
Doukhan, P. (1995). Mixing: Properties and examples. In Lecture notes in statistics (Vol. 85). Berlin: Springer.
Doukhan, P., Oppenheim, G., Taqqu, M.S. (eds) (2003) Theory and applications of long-range dependence. Birkhauser, Boston
Ibragimov, I. A., Has’minskii, R. Z. (1972). The asymptotic behavior of certain statistical estimates in the smooth case. I. Investigation of the likelihood ratio. (Russian) Teorija Verojatnostei i ee Primenenija, 17, 469–486.
Ibragimov, I. A., Has’minskii, R. Z. (1973). Asymptotic behavior of certain statistical estimates. II. Limit theorems for a posteriori density and for Bayesian estimates. (Russian) Teorija Verojatnostei i ee Primenenija, 18, 78–93.
Ibragimov I.A., Has’minskii R.Z. (1981) Statistical estimation: Asymptotic theory. Springer, New York
Kessler M. (1997) Estimation of diffusion processes from discrete observations. Scandinavian Journal of Statistics, Theory and Applications 24: 211–229
Kusuoka S., Yoshida N. (2000) Malliavin calculus, geometric mixing, and expansion of diffusion functionals. Probability Theory and Related Fields 116: 457–484
Kutoyants, Y. A. (1984). Parameter estimation for stochastic processes (B. L. S. Prakasa Rao, Ed., Trans.). Berlin: Herdermann.
Kutoyants Y. (1994) Identification of dynamical systems with small noise. Kluwer, Dordrecht
Kutoyants, Y. A. (1998). Statistical inference for spatial Poisson processes. In Lecture notes in statistics (Vol. 134). Berlin: Springer.
Kutoyants, Y. A. (2004). Statistical inference for ergodic diffusion processes. In Springer series in statistics. London: Springer.
Meyn, S. P., Tweedie, R. L. (1992). Stability of Markovian processes. I. Criteria for discrete-time chains. Advances in Applied Probability, 24, 542–574.
Meyn, S. P., Tweedie, R. L. (1993a). Stability of Markovian processes. II. Continuous-time processes and sampled chains. Advances in Applied Probability, 25, 487–517.
Meyn, S. P., Tweedie, R. L. (1993b). Stability of Markovian processes. III. Foster–Lyapunov criteria for continuous-time processes. Advances in Applied Probability, 25, 518–548.
Prakasa Rao B.L.S. (1983) Asymptotic theory for non-linear least square estimator for diffusion processes. Mathematische Operationsforschung und Statistik Series Statistics 14: 195–209
Prakasa-Rao B.L.S. (1988) Statistical inference from sampled data for stochastic processes. Contemporary Mathematics 80: 249–284
Rio, E. (1994). Inegalites de moments pour les suites stationnaires et fortement melangeantes. (French) [Moment inequalities for stationary strongly mixing sequences] Comptes Rendus de l’Académie des Sciences. Série I. Mathématique, 318, 355–360.
Sakamoto, Y., Yoshida, N. (1999). Higher order asymptotic expansions for a functional of a mixing process and applications to diffusion functionals (unpublished).
Sakamoto Y., Yoshida N. (2004) Asymptotic expansion formulas for functionals of epsilon—Markov processes with a mixing property. Annals of the Institute of Statistical Mathematics 56: 545–597
Shimizu, Y., Yoshida, N. (2002). Estimation of diffusion processes with jumps from discrete observations. Statistical Inference for Stochastic Processes (to appear).
Uchida M., Yoshida N. (2001) Information criteria in model selection for mixing processes. Statistical Inference for Stochastic Processes 4: 73–98
Varakin A.B., Veretennikov A.Y. (2002) On parameter estimation for “polynomial ergodic” Markov chains with polynomial growth loss functions. Markov Processes and Related Fields 8: 127–144
Veretennikov A.Y. (1987) Bounds for the mixing rate in the theory of stochastic equations. Theory of Probability and its Applications 32: 273–281
Veretennikov A.Y. (1997) On polynomial mixing bounds for stochastic differential equations. Stochastic Processes and their Applications 70: 115–127
Yoshida, N. (1988). On asymptotic mixed normality of the maximum likelihood estimator in a multidimensional diffusion process. Statistical theory and data analysis, II (Tokyo, 1986) (pp. 559–566). Amsterdam: North-Holland.
Yoshida N. (1990) Asymptotic behavior of M-estimator and related random field for diffusion process. Annals of the Institute of Statistical Mathematics 42(2): 221–251
Yoshida, N. (1992a). Asymptotic expansion for small diffusions via the theory of Malliavin–Watanabe. Probability Theory and Related Fields, 92, 275–311 (1992).
Yoshida N. (1992) Estimation for diffusion processes from discrete observation. Journal of Multivariate Analysis 41: 220–242
Yoshida N. (1993) Asymptotic expansion of Bayes estimators for small diffusions. Probability Theory and Related Fields 95: 429–450
Yoshida N. (1995) Malliavin calculus and asymptotic expansion for martingales. Probability Theory and Related Fields 109: 301–342
Yoshida N. (2004) Partial mixing and conditional Edgeworth expansion. Probability Theory and Related Fields 129: 559–624
Yoshida, N. (2005a). General M-estimation for stochastic differential equation with jumps by sampled data (in preparation).
Yoshida, N. (2005b). Asymptotic expansion for general M-estimation for stochastic differential equation with jumps. Zenkin Tenkai 2003, Hiroshima International University (in preparation).
Yoshida, N. (2006). Polynomial type large deviation inequalities and convergence of statistical random fields. Research Memorandum, The Institute of Statistical Mathematics.
Yoshida, N. (2007). Statistical random field with graded parameters and mighty convergence. In Workshop on Stochastic Analysis and Statistics held at ISM, February 15, 2007.
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This work was in part supported by JSPS Grants-in-Aid for Scientific Research No. 19340021, JST Basic Research Programs PRESTO, Cooperative Research Program of the Institute of Statistical Mathematics, and by 21st Century COE Program “Base for New Development of Mathematics to Science and Technology” and the Global COE Program “The research and training center for new development in mathematics”, Graduate School of Mathematical Sciences, University of Tokyo. The author thanks the referees for their valuable comments. Also he wishes to acknowledge the useful correspondence with Prof. Yury Kutoyants, Prof. Masayuki Uchida, Mr. T. Ogihara and Mr. H. Inatsugu.
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Yoshida, N. Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations. Ann Inst Stat Math 63, 431–479 (2011). https://doi.org/10.1007/s10463-009-0263-z
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DOI: https://doi.org/10.1007/s10463-009-0263-z