Bayesian estimation of a decreasing density

Abstract

Suppose ${\mathit{X}}_1,\dots ,{\mathit{X}}_{\mathit{n}}$ is a random sample from a bounded and decreasing density ${\mathit{f}}_0$ on $[0,\infty )$. We are interested in estimating such ${\mathit{f}}_0$, with special interest in ${\mathit{f}}_0(0)$. 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 (Electronic Journal of Statistics 8 (2014) 1380–1404). Several choices of base measure are numerically evaluated and compared. In a simulation various frequentist methods and a Bayesian estimator are compared. Finally, the Bayesian procedure is applied to current durations data described in Slama et al. (Human Reproduction 27 (2012) 1489–1498).