Abstract
Long-tailed distributions arise in many areas of the sciences. These distributions, however, suffer from the weakness of not having finite moments of all orders and this weakness has restricted their use. In this note, we introduce truncated versions of five of the most commonly known long-tailed distributions—which possess finite moments of all orders and could therefore be better models. Explicit expressions for the moments are derived for each of the truncated distributions. Several applications are illustrated using real data.
Similar content being viewed by others
References
Abate, J., Whitt, W.: Computing Laplace transforms for numerical inversion via continued fractions. INFORMS J. Comput. 11, 394–405 (1999)
Adler, R.J., Feldmann, R.E., Taqqu, M.S.: A Practical Guide to Heavy Tails. Birkhäuser, Boston (1998)
Bhowmick, K., Mukhopadhyay, A., Mitra, G.B.: Edgeworth series expansion of the truncated Cauchy function and its effectiveness in the study of atomic heterogeneity. Z. Kristallogr. 215, 718–726 (2000)
Crovella, M.E., Taqqu, M.S., Bestavros, A.: Heavy-tailed probability distributions in the world wide Web. In: Adler, R.J., Feldmann, R.E., Taqqu, M.S. (eds.) A Practical Guide to Heavy Tails, pp. 3–25. Birkhäuser, Boston (1998)
Fair, R.C.: Theory of extra-marital affairs. J. Political Econ. 86, 45–61 (1978)
Feldmann, A., Whitt, W.: Fitting mixtures of exponentials to long-tail distributions to analyze network performance models. Perform. Eval. 31, 963–976 (1998)
Gleeson, J.P.: Passive motion in dynamical disorder as a model for stock market prices. Phys. A Stat. Mech. Appl. 351, 523–550 (2005)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 6th edn. Academic Press, San Diego (2000)
Greene, W.H.: Econometric Analysis, 5th edn. Prentice Hall, New Jersey (2003)
Groot, R.D.: Consumers don’t play dice, influence of social networks and advertisements. Phys. A Stat. Mech. Appl. 363, 446–458 (2006)
Gupta, H.M., Campanha, J.R.: Power-law distribution in a learning process: competition, learning and natural selection. Phys. A Stat. Mech. Appl. 345, 267–274 (2005)
Harlow, D.G.: Applications of the Frechet distribution function. Int. J. Mater. Prod. Technol. 17, 482–495 (2002)
Jaroszewicz, S., Mariani, M.C., Ferraro, M.: Long correlations and truncated Levy walks applied to the study Latin-American market indices. Phys. A Stat. Mech. Appl. 355, 461–474 (2005)
Leland, W.E., Taqqu, M.S., Willinger, W., Wilson, D.V.: On the self-similar nature of Ethernet traffic. IEEE/ACM Trans. Netw. 2, 1–15 (1994)
Nadarajah, S., Kotz, S.: Skewed distributions generated by the normal kernel. Stat. Probab. Lett. 65, 269–277 (2003)
Paxson, V., Floyd, S.: Wide-area traffic: the failure of Poisson modeling. IEEE/ACM Trans. Netw. 3, 226–244 (1995)
Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series. Gordon and Breach Science Publishers, Amsterdam (1986). Volumes 1, 2 and 3
Shmueli, U.: Symmetry and composition dependent cumulative distribution of the normalized structure amplitude for use in intensity statistics. Acta Crystallogr. A 35, 282–286 (1979)
Shmueli, U., Wilson, A.J.C.: Effects of space group symmetry and atomic heterogeneity on intensity statistics. Acta Crystallogr. A 37, 342–353 (1981)
Wald, A.: Tests of statistical hypotheses concerning several parameters when the number of observations is large. Trans. Am. Math. Soc. 54, 426–483 (1943)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nadarajah, S. Some Truncated Distributions. Acta Appl Math 106, 105–123 (2009). https://doi.org/10.1007/s10440-008-9285-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-008-9285-4