Charrelation and Charm: Generic Statistics Incorporating Higher-Order Information

Classical Higher-Order Statistics (HOS, in the form of high-order cumulants) are a powerful tool in the context of multivariate statistical analysis, often entailing valuable statistical information beyond Second-Order statistics (SOS), albeit at the expense of increased computational and notational complexity and compromised statistical stability (in the sense that longer observation intervals might be required in order to fully realize the advantages of HOS over SOS). In this paper, we consider new generic tools, offering the structural simplicity and controllable statistical stability of SOS on the one hand, yet retaining higher-order statistical information on the other hand. While cumulants are related to high-order derivatives of the log characteristic function at the origin, our new tools are related to lower-order (first and second) derivatives away from the origin, at locations called processing-points, and are termed charmean (or charm, in short) and charrelation. The charm and charrelation coincide with the classical mean and covariance (resp.) when the processing-point approaches the origin, but can offer continuously tunable tradeoffs between statistical stability and information contents as the processing-point is dragged away from the origin. Our goal in this paper is to introduce the underlying mathematical-statistical concepts for the development and analysis of charm- and charrelation-based estimation. We derive explicit expressions for the asymptotic bias and variance of their sample-estimates, which in turn enable data-driven selection of the processing-points, so as to minimize the predicted mean square estimation error in a given problem-as we demonstrate in several simulation examples.