Concentration rate and consistency of the posterior distribution for selected priors under monotonicity constraints
Salomond, Jean-Bernard (2014), Concentration rate and consistency of the posterior distribution for selected priors under monotonicity constraints, Electronic Journal of Statistics, 8, 1, p. 1380-1404. 10.1214/14-EJS929
TypeArticle accepté pour publication ou publié
External document linkhttp://dx.doi.org/10.1214/14-EJS929
Journal nameElectronic Journal of Statistics
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Abstract (EN)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.
Subjects / KeywordsDensity estimation; Bayesian inference; concentration rate
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