A parametric family of Bayesian estimators for non-standard loss functions

This paper introduces a new parametric family of Bayesian estimators. As the estimation with non-standard loss functions can often only be stated as an optimization problem which has to be solved for each new observation, it is advantageous to use such a parametric family. We proof that many well known estimators are included in our family. Among them are the MMSE and MAP estimator as well as the optimal Bayesian estimator (OBE) under LINEX loss. By restricting the estimator to lie in this family, we split the estimation problem into two parts: In a first step, we have to find the best estimator with respect to the Bayes risk for a given loss function, which has to be done only once. The second step then calculates the estimate for a given observation. We demonstrate the usefulness of the proposed parametric family in an example.