Asymptotic expansions for perturbed systems on Wiener space: Maximum likelihood estimators
We consider stochastic systems with unknown parameters disturbed by white Gaussian noises or normal random variables. For many such systems the unknown parameters can be estimated consistently by certain statistical estimators when the disturbances become small and the stochastic system tends to the corresponding deterministic one. For instance, the maximum likelihood method and the Bayes method are available for diffusion processes with unknown parameters in their drifts when the diffusion coefficient is small. In this case, the maximum likelihood estimator and the Bayes estimator are consistent and efficient in the first order. As for higher order properties of estimators, they are known to be second-order efficient in a certain sense. This fact follows from their asymptotic expansions in consideration of a problem of hypothesis testing. Thus asymptotic expansions for estimators play an important role in higher order statistical inference. The purpose of this article is to derive asymptotic expansions for likelihood ratio statistics and maximum likelihood estimators of unknown parameters involved in a system slightly disturbed by white Gaussian noises or normal random variables.
scientific article; zbMATH DE number 889777