On conditions for linearity of optimal estimation

When is optimal estimation linear? It is well-known that, in the case of a Gaussian source contaminated with Gaussian noise, a linear estimator minimizes the mean square estimation error. This paper analyzes more generally the conditions for linearity of optimal estimators. Given a noise (or source) distribution, and a specified signal to noise ratio (SNR), we derive conditions for existence and uniqueness of a source (or noise) distribution that renders the L_p norm optimal estimator linear. We then show that, if the noise and source variances are equal, then the matching source is distributed identically to the noise. Moreover, we prove that the Gaussian source-channel pair is unique in that it is the only source-channel pair for which the MSE optimal estimator is linear at more than one SNR values.