Bernstein–von Mises theorem for linear functionals of the density
Rivoirard, Vincent; Rousseau, Judith (2012), Bernstein–von Mises theorem for linear functionals of the density, Annals of Statistics, 40, 3, p. 1489-1523. http://dx.doi.org/10.1214/12-AOS1004
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00405849
Journal nameAnnals of Statistics
Institute of Mathematical Statistics
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Abstract (EN)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.
Subjects / KeywordsBayesian nonparametric; rates of convergence; Bernstein–von Mises; adaptive estimation
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