Abstract
In this paper we define the class of matrix Mittag-Leffler distributions and study some of its properties. We show that it can be interpreted as a particular case of an inhomogeneous phase-type distribution with random scaling factor, and alternatively also as the absorption time of a semi-Markov process with Mittag-Leffler distributed interarrival times. We then identify this class and its power transforms as a remarkably parsimonious and versatile family for the modeling of heavy-tailed risks, which overcomes some disadvantages of other approaches like the problem of threshold selection in extreme value theory. We illustrate this point both on simulated data as well as on a set of real-life MTPL insurance data that were modeled differently in the past.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albrecher, H., Beirlant, J., Teugels, J.L.: Reinsurance: Actuarial and Statistical Aspects. Wiley, New York (2017)
Albrecher, H., Bladt, M.: Inhomogeneous phase–type distributions and heavy tails. J. Appl. Probab. 56(4), 1044–1064 (2019)
Asmussen, S., Nerman, O., Olsson, M.: Fitting phase-type distributions via the EM algorithm. Scandinavian Journal of Statistics, 419–441 (1996)
Asmussen, S., Albrecher, H.: Ruin probabilities, 2nd edn. World Scientific, Hackensack (2010)
Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.L.: Statistics of Extremes: Theory and Applications. Wiley, Chichester (2004)
Bladt, M., Nielsen, B.F., Samorodnitsky, G.: Calculation of ruin probabilities for a dense class of heavy tailed distributions. Scandinavian Actuarial Journal, 573–591 (2015)
Bladt, M., Nielsen, B.F.: Matrix-Exponential Distributions in Applied Probability. Springer, Berlin (2017)
Bladt, M., Albrecher, H., Beirlant, J.: Combined tail estimation using censored data and expert information. Preprint University of Lausanne. Scandinavian Actuarial Journal (2019a). https://doi.org/10.1080/03461238.2019.1694974
Bladt, M., Albrecher, H., Beirlant, J.: Trimming and threshold selection in extremes. ar**v:1903.07942 (2019b)
Bladt, M., Rojas-Nandayapa, L.: Fitting phase–type scale mixtures to heavy–tailed data and distributions. Extremes 21(2), 285–313 (2018)
Chikrii, A.A., Eidel’man, SD.: Generalized Mittag-Leffler matrix functions in game problems for evolutionary equations of fractional order. Cybern. Syst. Anal. 36 (3), 315–338 (2000)
Constantinescu, C.D., Ramirez, J.M., Zhu, W.R.: An application of fractional differential equations to risk theory. Finance and Stochastics 23, 1001–1024 (2019)
Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events, Volume 33 of Applications of Mathematics (New York). Springer, Berlin. For insurance and finance (1997)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher transcendental Functions, vol. III. McGraw-Hill Book Company, Inc., New York (1955). Based, in part, on notes left by Harry Bateman
Feller, W.: An introduction to probability theory and its applications, vol. II. Wiley, New York (1971)
Garrappa, R., Popolizio, M.: Computing the matrix Mittag-Leffler function with applications to fractional calculus. J. Sci. Comput. 77(1), 129–153 (2018)
Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-leffler functions, related topics and applications. Springer, Berlin (2014)
Haubold, H.J., Mathai, A.M., Saxena, R.K.: Mittag-Leffler functions and their applications. Journal of Applied Mathematics, Article ID 298628, pp. 51 (2011)
Jose, K.K., Uma, P., Lekshmi, V.S., Haubold, H.J.: Generalized Mittag-Leffler distributions and processes for applications in astrophysics and time series modeling. In: Proceedings of the Third UN/ESA/NASA Workshop on the International Heliophysical Year 2007 and basic space science, pp. 79–92. Springer, Berlin (2010)
Klugman, S.A., Panjer, H.H., Willmot, G.E.: Loss models: from data to decisions. Wiley, New York (2012)
Kozubowski, T.J.: Fractional moment estimation of Linnik and Mittag-Leffler parameters. Math. Comput. Modell. 34(9-11), 1023–1035 (2001)
Matychyn, I., Onyshchenko, V.: Matrix Mittag-Leffler function in fractional systems and its computation. Bullet. Polish Acad. Sci. Tech. Sci. 66(4), 495–500 (2018)
Mikosch, T.: Regular variation, subexponentiality and their applications in probability theory. Eurandom Report 99013 Eindhoven University of Technology (1999)
Mittag-Leffler, M.G.: Sopra la funzione Eα(x). Rend. Accad. Lincei 13(5), 3–5 (1904)
Nešlehová, J., Embrechts, P., Chavez-Demoulin, V.: Infinite mean models and the LDA for operational risk. J. Oper. Risk 1(1), 3–25 (2006)
Pigeon, M., Denuit, M.: Composite Lognormal-Pareto model with random threshold. Scand. Actuar. J., 177–192 (2011)
Pillai, R. N.: On Mittag-Leffler functions and related distributions. Ann. Inst. Statist. Math. 42(1), 157–161 (1990)
Wolfe, S.J.: On Moments of Probability Distribution Functions. In: Fractional Calculus and Its Applications, pp. 306–316. Springer, Berlin (1975)
Acknowledgments
H.A. acknowledges financial support from the Swiss National Science Foundation Project 200021_168993.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Albrecher, H., Bladt, M. & Bladt, M. Matrix Mittag–Leffler distributions and modeling heavy-tailed risks. Extremes 23, 425–450 (2020). https://doi.org/10.1007/s10687-020-00377-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10687-020-00377-0
Keywords
- Matrix distributions
- Mittag-Leffler functions
- Heavy tails
- Risk modeling
- Phase-type distributions
- Random scaling