  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).
