  Estimation of the mixing distribution under a general mixture model is a very
difficult problem, especially when the mixing distribution is assumed to have a
density. Predictive recursion (PR) is a fast, recursive algorithm for
nonparametric estimation of a mixing distribution/density in general mixture
models. However, the existing PR consistency results make rather strong
assumptions, some of which fail for a class of mixture models relevant for
monotone density estimation, namely, scale mixtures of uniform kernels. In this
paper, we develop new consistency results for PR under weaker conditions. Armed
with this new theory, we prove that PR is consistent for the scale mixture of
uniforms problem, and we show that the corresponding PR mixture density
estimator has very good practical performance compared to several existing
methods for monotone density estimation.
