Optimal Bayesian Estimation of a Regression Curve, a Conditional Density, and a Conditional Distribution

In this paper, several related estimation problems are addressed from a Bayesian point of view, and optimal estimators are obtained for each of them when some natural loss functions are considered. The problems considered are the estimation of a regression curve, a conditional distribution function, a conditional density, and even the conditional distribution itself. These problems are posed in a sufficiently general framework to cover continuous and discrete, univariate and multivariate, and parametric and nonparametric cases, without the need to use a specific prior distribution. The loss functions considered come naturally from the quadratic error loss function commonly used in estimating a real function of the unknown parameter. The cornerstone of these Bayes estimators is the posterior predictive distribution. Some examples are provided to illustrate the results.