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On the asymptotic behavior of the parameter estimators for some diffusion processes: application to neuronal models. (English) Zbl 1215.62085

Summary: We consider a sample \(\{T_n\}_{1\leq n\leq N}\) of i.i.d. times and we interpret each item as the first-passage time (FPT) of a diffusion process through a constant boundary. The problem is to estimate the parameters characterizing the underlying diffusion process through the experimentally observable FPT’s. Recently S. Ditlevsen and P. Lánský [Phys. Rev. E 71, 011907, 9 pp. (2005); ibid. 73, 061910, 9 pp. (2006)] have closed form estimators proposed for neurobiological applications. Here we study the asymptotic properties (consistency and asymptotic normality) of the class of moment type estimators for parameters of diffusion processes like those of Ditlevsen and Lánský. Furthermore, to make our results useful for application instances we establish upper bounds for the rate of convergence of the empirical distribution of each estimator to the normal density. Applications are also considered by means of simulated experiments in a neurobiological context.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62F12 Asymptotic properties of parametric estimators
60J60 Diffusion processes
62P10 Applications of statistics to biology and medical sciences; meta analysis
92C20 Neural biology
65C20 Probabilistic models, generic numerical methods in probability and statistics
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