L ¹ Estimation in Gaussian Noise: On the Optimality of Linear Estimators

Consider the problem of estimating a random variable X in Gaussian noise under L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> fidelity criteria. It is well-known that in the L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> setting, the optimal Bayesian estimator is given by the conditional median. The goal of this work is to characterize the set of prior distributions on X for which the conditional median corresponds to a linear estimator. This work shows that neither discrete nor compactly supported distributions can induce a linear conditional median. Moreover, under certain non-trivial restrictions on the set of allowed probability distributions, the Gaussian is shown to be the only solution that induces a linear conditional median.