An Orthogonality Principle for Select-Maximum Estimation of Exponential Variables

Motivated by multiple-description source coding with feedback, it was recently proposed to encode the one-sided exponential source X via K parallel channels, Y1, ... , YK, such that the error signals X−Yi, i=1,...,K, are one-sided exponential and mutually independent given X. Moreover, it was shown that the optimal estimator hat{Y} of the source X with respect to the one-sided error criterion, is simply given by the maximum of the outputs, i.e.,hat{Y} = max{Y1,..., YK}. In this paper, we show that the distribution of the resulting estimation error X - hat{Y}, is equivalent to that of the optimum noise in the backward test-channel of the one-sided exponential source, i.e., it is one-sided exponentially distributed and statistically independent of the joint output Y1,..., YK.