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

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  • Agustín G. Nogales

    (Departamento de Matemáticas, IMUEx, Universidad de Extremadura, 06006 Badajoz, Spain)

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. 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.

Suggested Citation

  • Agustín G. Nogales, 2022. "Optimal Bayesian Estimation of a Regression Curve, a Conditional Density, and a Conditional Distribution," Mathematics, MDPI, vol. 10(8), pages 1-22, April.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:8:p:1213-:d:789052
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    References listed on IDEAS

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    1. Ghosal,Subhashis & van der Vaart,Aad, 2017. "Fundamentals of Nonparametric Bayesian Inference," Cambridge Books, Cambridge University Press, number 9780521878265, January.
    2. Agustín G. Nogales, 2022. "On Bayesian estimation of densities and sampling distributions: The posterior predictive distribution as the Bayes estimator," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 76(2), pages 236-250, May.
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