Electronic Journal of Statistics

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

Jean-Bernard Salomond

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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.

Article information

Electron. J. Statist., Volume 8, Number 1 (2014), 1380-1404.

First available in Project Euclid: 20 August 2014

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Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62G20: Asymptotic properties

Density estimation Bayesian inference concentration rate


Salomond, Jean-Bernard. Concentration rate and consistency of the posterior distribution for selected priors under monotonicity constraints. Electron. J. Statist. 8 (2014), no. 1, 1380--1404. doi:10.1214/14-EJS929. https://projecteuclid.org/euclid.ejs/1408540291

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