Estimation of mixture densities for the classical Gaussian compound decision problem and their associated (empirical) Bayes rules is considered from two new perspectives. The first, motivated by Brown and Greenshtein, introduces a nonparametric maximum likelihood estimator of the mixture density...

Given a scalar random variable Y and a random vector X defined on the same probability space, the conditional distribution of Y given X can be represented by either the conditional distribution function or the conditional quantile function. To these equivalent representations correspond two...