Expectation Values and Variance Based on L^p-Norms

Abstract: This analysis introduces a generalization of the basic statistical concepts of expectation values and variance for non-Euclidean metrics induced by L^p-norms. The non-Euclidean L^p means are defined by exploiting the fundamental property of minimizing the L^p deviations that compose the L^p variance. These L^p expectation values embody a generic formal scheme of means characterization. Having the p-norm as a free parameter, both the L^p-normed expectation values and their variance are flexible to analyze new phenomena that cannot be described under the notions of classical statistics based on Euclidean norms. The new statistical approach provides insights into regression theory and Statistical Physics. Several illuminating examples are examined.