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Local Asymptotic Mixed Normality for Nonhomogeneous Diffusions

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Part of the Lecture Notes in Mathematics book series (LNM, volume 1923)

We study the asymptotic properties of various estimators of the parameter appearing nonlinearly in the nonhomogeneous drift coefficient of a functional stochastic differential equation when the corresponding solution process, called the diffusion type process, is observed over a continuous time interval [0, T]. We show that the maximum likelihood estimator, maximum probability estimator and regular Bayes estimators are strongly consistent and when suitably normalised, converge to a mixture of normal distribution and are locally asymptotically minimax in the Hajek-Le Cam sense as T → ∞ under some regularity conditions. Also we show that posterior distributions, suitably normalised and centered at the maximum likelihood estimator, converge to a mixture of normal distribution. Further, the maximum likelihood estimator and the regular Bayes estimators are asymptotically equivalent as T → ∞. We illustrate the results through the exponential memory Ornstein-Uhlenbeck process, the nonhomogeneous Ornstein-Uhlenbeck process and the Kalman- Bucy filter model where the limit distribution of the above estimators and the posteriors is shown to be Cauchy.

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© Springer-Verlag Berlin Heidelberg 2008

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