## Brazilian Journal of Probability and Statistics

### Fractional absolute moments of heavy tailed distributions

#### Abstract

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.

#### Article information

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

Dates
Accepted: January 2015
First available in Project Euclid: 31 March 2016

https://projecteuclid.org/euclid.bjps/1459429713

Digital Object Identifier
doi:10.1214/15-BJPS280

Mathematical Reviews number (MathSciNet)
MR3481104

Zentralblatt MATH identifier
1381.60039

#### Citation

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. https://projecteuclid.org/euclid.bjps/1459429713

#### References

• Adler, R. J., Feldman, R. E. and Taqqu, M. S., eds. (1998). A Practical Guide to Heavy Tails: Statistical Techniques and Applications. New York: Birkhäuser.
• Blattberg, R. and Sargent, T. (1971). Regression with non-Gaussian stable disturbances: Some sampling results. Econometrica 39, 501–510.
• Brown, B. M. (1970). Characteristic functions, moments, and the central limit theorem. The Annals of Mathematical Statistics 41, 658–664.
• Brown, B. M. (1972). Formulae for absolute moments. Journal of the Australian Mathematical Society 13, 104–106.
• Cline, D. B. H. and Brockwell, P. J. (1985). Linear prediction of ARMA processes with infinite variance. Stochastic Processes and Their Applications 19, 281–296.
• Cressie, N. and Borkent, M. (1986). The moment generating function has its moments. Journal of Statistical Planning and Inference 13, 337–344.
• Cressie, N., Davis, A. S., Folks, J. L. and Policello, G. E. (1981). The moment-generating function and negative integer moments. American Statistician 35, 148–150.
• Devroye, L. (1990). A note on Linnik’s distribution. Statistics and Probability Letters 9, 305–306.
• Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of Integrals, Series, and Products, 7th ed. San Diego, CA: Academic Press.
• Hardin, C. D., Jr., Samorodnitsky, G. and Taqqu, M. S. (1991). Nonlinear regression of stable random variables. The Annals of Applied Probability 1, 582–612.
• Hsu, P. L. (1951). Absolute moments and characteristic functions. Journal Chinese Mathematical Society (New Series) 1, 257–280.
• Kawata, T. (1972). Fourier Analysis in Probability Theory. New York: Academic Press.
• Kokoszka, P. S. (1996). Prediction of inifinite variance fractional ARIMA. Probability and Mathematical Statistics 16, 65–83.
• Kozubowski, T. J. (2001). Fractional moment estimation of Linnik and Mittag–Leffler parameters. Mathematical and Computer Modelling 34, 1023–1035.
• Kozubowski, T. J. and Meerschaert, M. M. (2009). A bivariate infinitely divisible distribution with exponential and Mittag–Leffler marginals. Statistics and Probability Letters 79, 1596–1601.
• Kozubowski, T. J., Podgórski, K. and Samorodnitsky, G. (1999). Tails of Lévy measure of geometric stable random variables. Extremes 1, 367–378.
• Laue, G. (1980). Remarks on the relation between fractional moments and fractional derivatives of characteristic functions. Journal of Applied Probability 17, 456–466.
• Laue, G. (1986). Results on moments of non-negative random variables. Sankhyā Series A 48, 299–314.
• Lim, S. C. and Teo, L. P. (2010). Analytic and asymptotic properties of multivariate generalized Linnik’s probability densities. The Journal of Fourier Analysis and Applications 16, 715–747.
• Lin, G. D. (1998). On the Mittag–Leffler distributions. Journal of Statistical Planning and Inference 74, 1–9.
• Linnik, Y. V. (1953). Linear forms and statistical criteria. I, II. Ukrainskij Matematicheskij Zhurnal 5, 207–243, 247–290 (in Russian). English translation in Selected Translation in Mathematical Statistics and Probability 3 (1962/1963), 1–90.
• Matsui, M. and Mikosch, T. (2010). Prediction in a Poisson cluster model. Journal of Applied Probability 47, 350–366.
• Matsui, M. and Takemura, A. (2006). Some improvements in numerical evaluation of symmetric stable density and its derivatives. Communications in Statistics. Theory and Methods 35, 149–172.
• Mikosch, T., Samorodnitsky, G. and Tafakori, L. (2013). Fractional moments of solutions to stochastic recurrence equations. Journal of Applied Probability 50, 969–982.
• Nguyen, T. T. (1995). Conditional distributions and characterizations of multivariate stable distribution. Journal of Multivariate Analysis 53, 181–193.
• Nolan, J. P. (2013). Multivariate elliptically contoured stable distributions: Theory and estimation. Computational Statistics 28, 2067–2089.
• Paolella, M. S. (2007). Intermediate Probability: A Computational Approach. Chichester: Wiley.
• Pinelis, I. (2011). Positive-part moments via the Fourier–Laplace transform. Journal of Theoretical Probability 24, 409–421.
• Podlubny, I. (1999). Fractional Differential Equations. San Diego, CA: Academic Press.
• Ramachandran, B. (1969). On characteristic functions and moments. Sankhyā Series A 31, 1–12.
• Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives: Theory and Applications. Yverdon: Gordon and Breach Science Publishers.
• Samorodnitsky, G. and Taqqu, M. S. (1991). Conditional moments and linear regression for stable random variables. Stochastic Processes and Their Applications 39, 183–199.
• Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Boca Raton, FL: Chapman and Hall/CRC.
• Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge Univ. Press.
• Shanbhag, D. N. and Sreehari, M. (1977). On certain self-decomposable distributions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 38, 217–222.
• von Bahr, B. (1965). On the convergence of moments in the central limit theorem. The Annals of Mathematical Statistics 36, 808–818.
• Wolfe, S. J. (1971). On moments of infinitely divisible distribution functions. The Annals of Mathematical Statistics 42, 2036–2043.
• Wolfe, S. J. (1973). On the local behavior of characteristic functions. Annals of Probability 1, 862–866.
• Wolfe, S. J. (1975a). On moments of probability distribution functions. In Fractional Calculus and Its Applications (B. Ross, ed.). Lecture Notes in Mathematics 457, 306–316. Berlin: Springer.
• Wolfe, S. J. (1975b). On derivatives of characteristic functions. Annals of Probability 3, 737–738.
• Wolfe, S. J. (1978). On the behavior of characteristic functions on the real line. Annals of Probability 6, 554–562.
• Zolotarev, V. M. (1986). One-Dimensional Stable Distributions. Translations of Mathematical Monographs 65. Providence, RI: Amer. Math. Soc.