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.

翻译：在一般混合物模型下对混合分布进行估计是一个非常困难的问题,特别是当混合分布被认为具有密度时。预测性循环(PR)是一种快速的、递归的算法,用于对一般混合物模型混合分布/密度进行非参数性估计。然而,现有的PR一致性结果得出了相当强烈的假设,其中有些与单体密度估计相关的混合物模型类别,即统一内核的比重混合物,未能实现。在本文中,我们在较弱的条件下为PR制定了新的一致性结果。根据这一新理论,我们证明PR与制服问题的规模混合是一致的,我们表明相应的PR混合物密度估计值与单体密度估计的几种现有方法相比,具有非常好的实际性能。