Abstract
In this paper we prove the weak consistency and the asymptotic normality of the maximum likelihood estimation based on discrete observations ofn independent Gaussian Markov processes. The Ornstein Uhlenbeck process is a special Gaussian Markov process. We derive asymptotic simultaneous confidence regions for the parameters of the Ornstein Uhlenbeck process as an application.
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References
Basawa IV, Prakasa Rao BLS (1980) Statistical inference for stochastic processes. Academic Press, London, New York, Toronto, Sydney, San Francisco
Billingsley P (1961) Statistical inference for Markov processes. University of Chicago Press, Chicago
Dacunha-Castelle D, Florens-Zmirou D (1986) Estimation of the coefficients of a diffusion from discrete observations. Stochastics 19:263–284
Dacunha-Castelle D, Duflo M (1982) Probability and statistics, 2. Springer-Verlag New York
Doob JL (1949) Heuristic approach to the Kolmogorov-Smirnov-theorems. Ann Math Stat 20:393–403
Florens-Zmirou D (1989) Approximate discrete-time schemes for statistics of diffusions processes. Statistics 20, 4:547–557
Gänssler P (1972) Note on minimum contrast estimates for Markov processes. Metrika 19:115–130
Gänssler P, Stute W (1977) Wahrscheinlichkeitstheorie. Springer-Verlag, Berlin, Heidelberg, New York
Gaschler B (1994a) Parameterschätzungen bei Gauß-Markov-Prozessen. Dissertation, Otto-von-Guericke-Universität, Magdeburg
Gaschler B (1994b) On parameter transformation and asymptotic simultaneous confidence estimation for the Ornstein Uhlenbeck process from discrete observations. Statistical Papers (accepted)
Genon-Catalot V (1990) Maximum-contrast estimation for diffusion processes from discrete observations. Statistics 21, 1:99–116
Ibragimov IA, Has’minskij RZ (1981) Statistical estimation, asymptotic theory. Springer-Verlag Berlin
Kahle W (1994) Simultaneous confidence regions for the parameters of damage processes. Statistical Papers 35:27–41
Kahle W, Liese F (1993) Consistency and asymptotic normality of minimum contrast estimation in renewal processes. RoMaKo (accepted)
Mehr CB, McFadden JA (1965) Certain properties of Gaussian processes and their first passage times. J Roy Stat Soc 27:505–522
Pfanzagl J (1969) On the measurability and consistency of minimum contrast estimates. Metrika 14:249–276
Roussas G (1965) Extension to Markov processes of a result by A. Wald about the consistency of the maximum likelihood estimate. Z Wahrscheinlichkeitstheorie 4:69–73
Roussas G (1968) Asymptotic normality of the maximum likelihood estimate in Markov processes. Metrika 14:62–70
Serfling RJ (1980) Approximation theorems of mathematical statistics. John Wiley & Sons, New York, Chichester, Brisbane, Toronto
Witting H, Nölle G (1970) Angewandte Mathematische Statistik. Optimale finite und asymptotische Verfahren, B. G. Teubner, Stuttgart
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Gaschler, B. Consistency and asymptotic normality of maximum likelihood estimation for Gaussian Markov processes from discrete observations. Metrika 43, 69–90 (1996). https://doi.org/10.1007/BF02613898
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DOI: https://doi.org/10.1007/BF02613898