6533b871fe1ef96bd12d1b81
RESEARCH PRODUCT
Fractional calculus approach to the statistical characterization of random variables and vectors
Giulio CottoneRalf MetzlerMario Di Paolasubject
Statistics and ProbabilityMellin transformStatistical Mechanics (cond-mat.stat-mech)Characteristic function (probability theory)Multivariate distributionMultivariate random variableMathematical analysisFOS: Physical sciencesMoment-generating functionCondensed Matter PhysicsFractional calculusFractional and complex moments; Multivariate distributions; Power-law tails; Inverse Mellin transformFractional and complex momentIngenieurwissenschaftenApplied mathematicsddc:620Inverse Mellin transformSettore ICAR/08 - Scienza Delle CostruzioniRandom variableCondensed Matter - Statistical MechanicsMathematicsInteger (computer science)Taylor expansions for the moments of functions of random variablesPower-law taildescription
Fractional moments have been investigated by many authors to represent the density of univariate and bivariate random variables in different contexts. Fractional moments are indeed important when the density of the random variable has inverse power-law tails and, consequently, it lacks integer order moments. In this paper, starting from the Mellin transform of the characteristic function and by fractional calculus method we present a new perspective on the statistics of random variables. Introducing the class of complex moments, that include both integer and fractional moments, we show that every random variable can be represented within this approach, even if its integer moments diverge. Applications to the statistical characterization of raw data and in the representation of both random variables and vectors are provided, showing that the good numerical convergence makes the proposed approach a good and reliable tool also for practical data analysis.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2009-11-17 |