Bayesian estimation of a decreasing density

TL;DR

This paper develops a Bayesian method for estimating decreasing densities, especially at zero, using Dirichlet process mixtures, and compares it with other estimators through simulations and real data application.

Contribution

It introduces a Bayesian estimator for decreasing densities based on Dirichlet process mixtures and derives its posterior contraction rates at zero.

Findings

Bayesian estimator achieves consistent estimation at zero.

Simulation studies show competitive performance compared to existing methods.

Application to durations data demonstrates practical utility.

Abstract

Suppose $X_1,\dots, X_n$ is a random sample from a bounded and decreasing density $f_0$ on $[0,\infty)$. We are interested in estimating such $f_0$, with special interest in $f_0(0)$. This problem is encountered in various statistical applications and has gained quite some attention in the statistical literature. It is well known that the maximum likelihood estimator is inconsistent at zero. This has led several authors to propose alternative estimators which are consistent. As any decreasing density can be represented as a scale mixture of uniform densities, a Bayesian estimator is obtained by endowing the mixture distribution with the Dirichlet process prior. Assuming this prior, we derive contraction rates of the posterior density at zero by carefully revising arguments presented in Salomond (2014). Various methods for estimating the density are compared using a simulation study. We apply the Bayesian procedure to the current durations data described in Keiding et al.(2012).