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June 2012 Bernstein–von Mises theorem for linear functionals of the density
Vincent Rivoirard, Judith Rousseau
Ann. Statist. 40(3): 1489-1523 (June 2012). DOI: 10.1214/12-AOS1004


In this paper, we study the asymptotic posterior distribution of linear functionals of the density by deriving general conditions to obtain a semi-parametric 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.


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Vincent Rivoirard. Judith Rousseau. "Bernstein–von Mises theorem for linear functionals of the density." Ann. Statist. 40 (3) 1489 - 1523, June 2012.


Published: June 2012
First available in Project Euclid: 5 September 2012

zbMATH: 1257.62036
MathSciNet: MR3015033
Digital Object Identifier: 10.1214/12-AOS1004

Primary: 62G07 , 62G20

Keywords: adaptive estimation , Bayesian nonparametric , Bernstein–Von Mises , rates of convergence

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 3 • June 2012
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