Asymptotic Normality of Posterior Distributions for Exponential Families when the Number of Parameters Tends to Infinity

Exponential families arise naturally in statistical modelling and the maximum likelihood estimate (MLE) is consistent and asymptotically normal for these models [Berk [2]]. In practice, often one needs to consider models with a large number of parameters, particularly if the sample size is large; see Huber [14], Haberman [13] and Portnoy [18 21]. One may also think that the true model can only be approximated by a finite dimensional parametric model and the quality of the approximation improves with the dimension. In other words, we let the dimension of the parameter space grow with the sample size. Usual asymptotics of fixed dimension do not justify the large sample approximations in these situations and one needs more delicate results paying special attention to the increasing dimension. Consistency and asymptotic normality of the MLE in exponential families with an increasing number of parameters were established by Portnoy [21] under some conditions on the growth rate of the dimension of the parameter space. In this paper, we show that the doi:10.1006 jmva.1999.1874, available online at http: www.idealibrary.com on