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

A. G. Nogales

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arXiv · 2110.13427

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

A. G. Nogales

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Abstract

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. Namely, we are interested in estimating a regression curve. Simultaneously, the estimation problems of a conditional distribution function, or a conditional density, or even the conditional distribution itself, are considered. All these problems are posed in a sufficiently general framework to cover continuous and discrete, univariate and multivariate, parametric and non-parametric cases, without the need to use a specific prior distribution. The loss functions considered come naturally from the quadratic error loss function comonly used in estimating a real function of the unknown parameter. The cornerstone of the mentioned Bayes estimators is the posterior predictive distribution. Some examples are provided to illustrate these results.

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A. G. Nogales. 2021-10-26. Optimal Bayesian Estimation of a Regression Curve, a Conditional Density and a Conditional Distribution. https://arxiv.org/abs/2110.13427

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