Concentration rate and consistency of the posterior distribution for selected priors under monotonicity constraints

Jean-Bernard Salomond (1, 2)

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

In this paper, we consider the well known problem of estimating a density function under qualitative assumptions. More precisely, we estimate monotone non-increasing densities in a Bayesian setting and derive concentration rate for the posterior distribution for a Dirichlet process and finite mixture prior. We prove that the posterior distribution based on both priors concentrates at the rate (n/log(n))−1/3, which is the minimax rate of estimation up to a log(n) factor. We also study the behaviour of the posterior for the point-wise loss at any fixed point of the support of the density and for the sup-norm. We prove that the posterior distribution is consistent for both loss functions.

Mots clés: Density estimation Bayesian inference concentration rate

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

Jean-Bernard Salomond. Concentration rate and consistency of the posterior distribution for selected priors under monotonicity constraints. Electronic Journal of Statistics , 2014, 8 (1), http://projecteuclid.org/euclid.ejs/1408540291. ⟨10.1214/14-EJS929⟩. ⟨hal-01065564⟩