Fractional absolute moments of heavy tailed distributions

TL;DR

This paper introduces methods to calculate fractional absolute moments for heavy tailed distributions using fractional differentiation, providing analytical formulas and applications in prediction and estimation errors.

Contribution

It presents new formulas for fractional absolute moments of heavy tailed distributions using fractional differentiation techniques.

Findings

Derived analytical expressions for fractional moments of heavy tailed distributions.

Applied formulas to evaluate prediction and parameter estimation errors.

Demonstrated methods with several examples of heavy tailed distributions.

Abstract

Several convenient methods for calculation of fractional absolute moments are given with application to heavy tailed distributions. We use techniques of fractional differentiation to obtain formulae for $E[|X-\mu|^\gamma]$ with $1<\gamma<2$ and $\mu\in\mathbb{R}$. The main focus is on heavy tailed distributions, several examples are given with analytical expressions of fractional absolute moments. As applications, we calculate the fractional moment errors for both prediction and parameter estimation problems.