Title: Bayesian estimation of a decreasing density
Authors: Lixue Pang - Delft University of Technology (Netherlands) [presenting]
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 $f_0$, 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. We also address computational aspects of the problem and show how draws from the posterior can be obtained using Gibbs sampling. By a simulation study, we compare the behavior of various proposed methods for estimating $f_0(0)$. We further apply the algorithm to current duration data, where we construct pointwise credible regions for the density and distribution functions.