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2014 Concentration rate and consistency of the posterior distribution for selected priors under monotonicity constraints
Jean-Bernard Salomond
Electron. J. Statist. 8(1): 1380-1404 (2014). DOI: 10.1214/14-EJS929

Abstract

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.

Citation

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Jean-Bernard Salomond. "Concentration rate and consistency of the posterior distribution for selected priors under monotonicity constraints." Electron. J. Statist. 8 (1) 1380 - 1404, 2014. https://doi.org/10.1214/14-EJS929

Information

Published: 2014
First available in Project Euclid: 20 August 2014

zbMATH: 1298.62064
MathSciNet: MR3263126
Digital Object Identifier: 10.1214/14-EJS929

Subjects:
Primary: 62G07, 62G20

Rights: Copyright © 2014 The Institute of Mathematical Statistics and the Bernoulli Society

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Vol.8 • No. 1 • 2014
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