Consistency of the Bayes Estimator of a Regression Curve

TL;DR

This paper proves the strong consistency of the Bayes estimator for a regression curve under the $L^1$-squared loss and shows the convergence of its Bayes risk to zero for both $L^1$ and $L^1$-squared losses.

Contribution

It establishes the strong consistency and risk convergence of the Bayes estimator of a regression curve for specific loss functions.

Findings

Bayes estimator is strongly consistent under $L^1$-squared loss.

Bayes risk converges to zero for both $L^1$ and $L^1$-squared losses.

Results apply to the posterior predictive distribution.

Abstract

Strong consistency of the Bayes estimator of a regression curve for the $L^1$-squared loss function is proved. It is also shown the convergence to 0 of the Bayes risk of this estimator both for the $L^1$ and $L^1$-squared loss functions. The Bayes estimator of a regression curve is the regression curve with respect to the posterior predictive distribution.