Fractional absolute moments of heavy tailed distributions

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.