THE SEMIPARAMETRIC BERNSTEIN-VON MISES THEOREM

• 成果类型: Article
• 署名作者: Bickel, P. J.; Kleijn, B. J. K.
• 署名单位: University of California System; University of California Berkeley; University of Amsterdam
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/11-AOS921
• 发表日期: 2012
• 页码: 206-237
• 关键词: Nonparametric regression; rates; distributions; Consistency; CONVERGENCE; likelihood

摘要

In a smooth semiparametric estimation problem, the marginal posterior for the parameter of interest is expected to be asymptotically normal and satisfy frequentist criteria of optimality if the model is endowed with a suitable prior. It is shown that, under certain straightforward and interpretable conditions, the assertion of Le Cam's acclaimed, but strictly parametric, Bernstein-von Mises theorem [Univ. California Publ. Statist. 1 (1953) 277-329] holds in the semiparametric situation as well. As a consequence, Bayesian point-estimators achieve efficiency, for example, in the sense of Hajek's convolution theorem [Z. Wahrsch. Verw. Gebiete 14 (1970) 323-330]. The model is required to satisfy differentiability and metric entropy conditions, while the nuisance prior must assign nonzero mass to certain Kullback-Leibler neighborhoods [Ghosal, Ghosh and van der Vaart Ann. Statist. 28 (2000) 500-531]. In addition, the marginal posterior is required to converge at parametric rate, which appear to be the most stringent condition in examples. The results are applied to estimation of the linear coefficient in partial linear regression, with a Gaussian prior on a smoothness class for the nuisance.