Zero-Order Statistics: A Mathematical Framework for the Processing and Characterization of Very Impulsive Signals

Impulsive or heavy-tailed processes with infinite variance appear naturally in a variety of practical problems that include wireless communications, teletraffic, hydrology, geology, and economics. Most signal processing and statistical methods available in the literature have been designed under the assumption that the processes possess finite variance, and they usually break down in the presence of infinite variance. Although methods based on fractional lower-order statistics (FLOS) have proven successful in dealing with infinite variance processes, they fail in general when the noise distribution has very heavy algebraic tails. In this paper, we introduce a new approach to statistical moment characterization which is well defined over all processes with algebraic or lighter tails. Unlike FLOS, these zero-order statistics (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 foundation of ZOS. 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 analyze the properties of the new parameters, and derive a ZOS framework for location estimation that gives rise to a novel mode-type estimator with important optimality properties under very impulsive noise. Several simulations are presented to illustrate the potential of ZOS methods