BERNSTEIN-VON MISES THEOREM FOR LINEAR FUNCTIONALS OF THE DENSITY

• 成果类型: Article
• 署名作者: Rivoirard, Vincent; Rousseau, Judith
• 署名单位: Universite PSL; Universite Paris-Dauphine; Institut Polytechnique de Paris; ENSAE Paris
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/12-AOS1004
• 发表日期: 2012
• 页码: 1489-1523
• 关键词: CONVERGENCE-RATES; posterior distributions; Asymptotic Normality

摘要

In this paper, we study the asymptotic posterior distribution of linear functionals of the density by deriving general conditions to obtain a semiparametric version of the Bernstein-von Mises theorem. The special case of the cumulative distributive function, evaluated at a specific point, is widely considered. In particular, we show that for infinite-dimensional exponential families, under quite general assumptions, the asymptotic posterior distribution of the functional can be either Gaussian or a mixture of Gaussian distributions with different centering points. This illustrates the positive, but also the negative, phenomena that can occur in the study of Bernstein-von Mises results.