Bernstein Von Mises Theorem for linear functionals of the density

Vincent Rivoirard (1) , Judith Rousseau (1, 2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 CREST - Centre de Recherche en Économie et Statistique

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 for the study of Bernstein-von Mises results.

Mots clés: Bayesian nonparametric; Bernstein Von Mises; semiparametric inference; adaptive estimation; wavelet

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Vincent Rivoirard, Judith Rousseau. Bernstein Von Mises Theorem for linear functionals of the density. Annals of Statistics, 2012, 40, pp.1489-1523. ⟨10.1214/12-AOS1004SUPP⟩. ⟨hal-00405849v2⟩