On Asymptotic Properties of Bayes Type Estimators with General Loss Functions

We study asymptotic behaviors of Bayes type estimators and give sufficient conditions to obtain asymptotic limit distribution of estimation error. We assume polynomial type large deviation inequalities and prove asymptotic equivalence of estimation error of Bayes type estimator and that of M-estimator by the virtue of Ibragimov-Has'minskii's theory. The results can be applied to several statistical models of diffusion processes and jump diffusion processes. In this paper, we focus on application to a statistical model of an ergodic diffusion process and give asymptotic normality and convergence of moments of the Bayes type estimator with a general loss function.