A Bernstein-von Mises theorem for smooth functionals in semiparametric models

TL;DR

This paper establishes a Bernstein-von Mises theorem for smooth functionals in semiparametric models, providing new tools for bias handling and applying to diverse statistical problems including density estimation and autoregressive models.

Contribution

It develops a general BvM theorem for semiparametric functionals, addressing bias in nonlinear and low-regularity cases, and systematically studies priors in density estimation.

Findings

BvM theorem proven for various semiparametric models

New bias handling tools for nonlinear functionals

Systematic analysis of priors in density estimation

Abstract

A Bernstein-von Mises theorem is derived for general semiparametric functionals. The result is applied to a variety of semiparametric problems in i.i.d. and non-i.i.d. situations. In particular, new tools are developed to handle semiparametric bias, in particular for nonlinear functionals and in cases where regularity is possibly low. Examples include the squared $L^2$-norm in Gaussian white noise, nonlinear functionals in density estimation, as well as functionals in autoregressive models. For density estimation, a systematic study of BvM results for two important classes of priors is provided, namely random histograms and Gaussian process priors.