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, Y_{1}, \ldots, Y_{K}, such that the error signals X-Y_{i}, i=1, \ldots, 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\{Y_{1},\ldots, Y_{K}\}. 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 Y_{1}, \ldots, Y_{K}.