On the Kullback-Leibler Distance and the Mean Square Distortion of Mismatched Distributed Quantizers

We formulate a problem considering the implications of conditional mean estimation based on unknown or approximated source statistics with respect to the mean squared error of the estimates. Since we are specially interested in a solution for a multiterminal quantization problem where the statistical properties of the data to be estimated might play a key role towards finding a mathematical solution, we formulate the problem based on this concrete scenario without compromising a more general treatment of the problem. Specifically, we are interested in establishing a link between the mean squared error and the mismatch of used probability density function measured by the Kullback-Leibler distance.