  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.
