Brazilian Journal of Probability and Statistics

Fractional absolute moments of heavy tailed distributions

Muneya Matsui and Zbyněk Pawlas

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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 $\mathbb{E} [\vert X-\mu\vert^{\gamma}]$ with $1<\gamma<2$ and $\mu\in\mathbb{R}$. We review techniques of fractional differentiation of Laplace transforms and characteristic functions. Several examples are given with analytical expressions of $\mathbb{E} [\vert X-\mu\vert^{\gamma}]$. We also evaluate the fractional moment errors for both prediction and parameter estimation problems.

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Braz. J. Probab. Stat., Volume 30, Number 2 (2016), 272-298.

Received: December 2014
Accepted: January 2015
First available in Project Euclid: 31 March 2016

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Fractional absolute moments fractional derivatives heavy tailed distributions characteristic functions infinitely divisible distributions


Matsui, Muneya; Pawlas, Zbyněk. Fractional absolute moments of heavy tailed distributions. Braz. J. Probab. Stat. 30 (2016), no. 2, 272--298. doi:10.1214/15-BJPS280.

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