Fractional absolute moments of heavy tailed distributions

Several convenient methods for calculation of fractional absolute moments are given with application to heavy tailed distributions. Our main focus is on an infinite variance case with finite mean, that is, we are interested in formulae for E [vertical bar X - mu vertical bar(gamma)] with 1 < gamma < 2 and mu is an element of R. We review techniques of fractional differentiation of Laplace transforms and characteristic functions. Several examples are given with analytical expressions of E [vertical bar X - mu vertical bar gamma]. We also evaluate the fractional moment errors for both prediction and parameter estimation problems.