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Bayes estimation with asymmetrical cost functions (Corresp.)

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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} wheref_1(cdot), f_2(cdot)are both twice-differentiable convex upward positive functions on[0, infty]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 functionsf_1(cdot), f_2(cdot), the unbiased Bayes estimate is shown to be the conditional mean.

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Information Theory, IEEE Transactions on  (Volume:21 ,  Issue: 1 )