In this article, we introduce mixture representations for likelihood ratio ordered distributions. Essentially, the ratio of two probability densities, or mass functions, is monotone if and only if one can be expressed as a mixture of one-sided truncations of the other. To illustrate the practical value of the mixture representations, we address the problem of density estimation for likelihood ratio ordered distributions. In particular, we propose a nonparametric Bayesian solution which takes advantage of the mixture representations. The prior distribution is constructed from Dirichlet process mixtures and has large support on the space of pairs of densities satisfying the monotone ratio constraint. Posterior consistency holds under reasonable conditions on the prior specification and the true unknown densities. To our knowledge, this is the first posterior consistency result in the literature on order constrained inference. With a simple modification to the prior distribution, we can test the equality of two distributions against the alternative of likelihood ratio ordering. We develop a Markov chain Monte Carlo algorithm for posterior inference and demonstrate the method in a biomedical application.

翻译：本文针对似然比有序分布提出了混合表示方法。本质上，两个概率密度函数（或质量函数）之比具有单调性的充要条件是：其中一个分布可以表示为另一个分布单侧截断的混合。为阐明混合表示的实际应用价值，我们解决了似然比有序分布的密度估计问题。特别地，我们提出了一种利用混合表示的优势的非参数贝叶斯方法。先验分布基于狄利克雷过程混合构建，并在满足单调比约束的密度对空间上具有大支撑集。在先验设定和真实未知密度的合理条件下，后验一致性成立。据我们所知，这是有序约束推断文献中首个后验一致性结果。通过对先验分布进行简单修改，我们还可以检验两个分布相等性假设（备择假设为似然比序关系）。我们开发了用于后验推断的马尔可夫链蒙特卡洛算法，并在生物医学应用中验证了该方法。