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 $Y_{1}, \ldots, Y_{K}$.