We consider the estimation of unknown parameters in the drift and diffusion coefficients of a onedimensional ergodic diffusion X when the observation Y is a discrete sampling of X with an additive noise, at times iδ, i = 1, . . . , N. Assuming that the sampling interval tends to 0 while the total length-time interval tends to infinity, we prove limit theorems for functionals associated with the observations, based on local means of the sample. We apply these results to obtain a contrast function. The associated minimum contrast estimators are shown to be consistent. Some examples are discussed with numerical simulations.
In this article, general estimating functions for ergodic diffusions sampled at high frequency with noisy observations are presented. The theory is formulated in term of approximate martingale estimating functions based on local means of the observations, and simple conditions are given for rate optimality. The estimation of diffusion parameter is faster that the estimation of drift parameter, and the rate of convergence in the Central Limit Theorem is classical for the drift parameter but not classical for the diffusion parameter. The link with specific minimum contrast estimators is established, as an example.
We consider a bidimensional Ornstein-Uhlenbeck process to describe the tissue microvascularization in anti-cancer therapy. Data are discrete, partial and noisy observations of this stochastic differential equation (SDE). Our aim is to estimate the SDE parameters. We use the main advantage of a one-dimensional observation to obtain an easy way to compute the exact likelihood using the Kalman filter recursion, which allows to implement an easy numerical maximization of the likelihood. Furthermore, we establish the link between the observations and an ARMA process and we deduce the asymptotic properties of the maximum likelihood estimator. We show that this ARMA property can be generalized to a higher dimensional underlying Ornstein-Uhlenbeck diffusion. We compare this estimator with the one obtained by the well-known expectation maximization algorithm on simulated data. Our estimation methods can be directly applied to other biological contexts such as drug pharmacokinetics or hormone secretions. Copyright (c) 2010 Board of the Foundation of the Scandinavian Journal of Statistics.
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