The semiparametric Bernstein-von Mises theorem for models with symmetric error

Abstract

In a smooth semiparametric model, the marginal posterior distribution of the finite dimensional parameter of interest is expected to be asymptotically equivalent to the sampling distribution of frequentist's efficient estimators. This is the assertion of the so-called Bernstein-von Mises theorem, and recently, it has been proved in many interesting semiparametric models. In this thesis, we consider the semiparametric Bernstein-von Mises theorem in some models which have symmetric errors. The simplest example of these models is the symmetric location model that has 1-dimensional location parameter and unknown symmetric error. Also, the linear regression and random effects models are included provided the error distribution is symmetric. The condition required for nonparametric priors on the error distribution is very mild, and the most well-known Dirichlet process mixture of normals works well. As a consequence, Bayes estimators in these models satisfy frequentist criteria of optimality such as Hajek-Le Cam convolution theorem. The proof of the main result requires that the expected log likelihood ratio has a certain quadratic expansion, which is a special property of symmetric densities. One of the main contribution of this thesis is to provide an efficient estimator of regression coefficients in the random effects model, in which it is unknown to estimate the coefficients efficiently because the full likelihood inference is difficult. Our theorems imply that the posterior mean or median is efficient, and the result from numerical studies also shows the superiority of Bayes estimators. For practical use of our main results, efficient Gibbs sampler algorithms based on symmetrized Dirichlet process mixtures are provided.