Bayes estimation with asymmetrical cost functions (Corresp.)

It is known that under certain restrictions on the posterior density and assigned cost function, the Bayes estimate of a random parameter is the conditional mean. The restrictions on the cost function are that it must be a symmetric convex upward function of the difference between the parameter and the estimate. In this correspondence, asymmetrical cost functions of the following form are examined: begin{equation} C(a, hat{a})= begin{cases} f_1(a- hat{a}),& a geq hat{a} \\ f_2(hat{a}- a),& a < hat{a} end{cases} end{equation} where are both twice-differentiable convex upward positive functions on that intersect the origin. It is shown that for posterior densities satisfying a certain symmetry condition, the biased Bayes estimate is a generalized median. Furthermore, for linear polynomial functions , the unbiased Bayes estimate is shown to be the conditional mean.