The asymptotic equivalence of Bayes and maximum likelihood estimation

Let (X, ) be a measurable space, [Theta] [subset, double equals] an open interval and P[Omega] [short parallel] , [Omega] [epsilon] [Theta], a family of probability measures fulfilling certain regularity conditions. Let [Omega]n be the maximum likelihood estimate for the sample size n. Let [lambda] be a prior distribution on [Theta] and let be the posterior distribution for the sample size n given . denotes a loss function fulfilling certain regularity conditions and Tn denotes the Bayes estimate relative to [lambda] and L for the sample size n. It is proved that for every compact K [subset, double equals] [Theta] there exists cK >= 0 such that This theorem improves results of Bickel and Yahav [3], and Ibragimov and Has'minskii [4], as far as the speed of convergence is concerned.