Zero - Order Statistics : A Signal Processing Framework for VeryImpulsive Processes

Impulsive or heavy-tailed processes with innnite variance appear naturally in a variety of practical problems that include wireless communications, teletraac, hidrology, geology and economics. Most signal processing and statistical methods available in the literature have been designed under the assumption that the processes possess nite variance, and they usually breakdown in the presence of innnite variance. Although methods based on fractional lower-order statistics (FLOS) have proven successful in dealing with innnite variance processes, they fail in general when the noise distribution has very heavy algebraic tails. In this paper we introduce the foundation of a new theory of statistics which is well deened over all processes with algebraic or lighter tails. Unlike FLOS, these zero-order statistics or ZOS, as we will call them, provide a common ground for the analysis of basically any distribution of practical use known today. Three new parameters, namely the geometric power, the zero-order location and the zero-order dispersion, constitute the underpinnings of ZOS theory. They play roles similar to those played by the power, the expected value and the standard deviation, in the theory of second-order processes. We discuss several important properties of the new parameters, and derive a ZOS framework for location estimation that gives rise to the discovery of a novel mode-type estimator with optimality properties under very impulsive noise. We also show the intimate relation between ZOS and FLOS. Given the limitations of signal processing methods based on second-order and fractional lower-order statistics, ZOS are an attractive alternative for signal processing applications in which innnite variance processes arise. All gures, simulations and source code utilized in this paper are reproducible and freely accessible in the Internet at