Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-15T17:26:36.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic tail behaviour of phase-type scale mixture distributions

Published online by Cambridge University Press:  29 August 2017

Leonardo Rojas-Nandayapa  and
Wangyue Xie
Leonardo Rojas-Nandayapa
Affiliation:
School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia
Wangyue Xie*
Affiliation:
School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia
*
*Correspondence to: Wangyue Xie, School of Mathematics and Physics, The University of Queensland, St Lucia, Brisbane, QLD 4072, Australia. Tel: +61 424494412; E-mail: w.xie1@uq.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider phase-type scale mixture distributions which correspond to distributions of a product of two independent random variables: a phase-type random variable Y and a non-negative but otherwise arbitrary random variable S called the scaling random variable. We investigate conditions for such a class of distributions to be either light- or heavy-tailed, we explore subexponentiality and determine their maximum domains of attraction. Particular focus is given to phase-type scale mixture distributions where the scaling random variable S has discrete support – such a class of distributions has been recently used in risk applications to approximate heavy-tailed distributions. Our results are complemented with several examples.

Keywords

Phase-typeDiscrete scale mixturesHeavy-tailedMaximum domain of attractionRuin probability
Type
Paper
Information
Annals of Actuarial Science , Volume 12 , Issue 2 , September 2018 , pp. 412 - 432
Copyright
© Institute and Faculty of Actuaries 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arendarczyk, M. & Dębicki, K. (2011). Asymptotics of supremum distribution of a Gaussian process over a Weibullian time. Bernoulli, 17(1), 194–210.Google Scholar
Asimit, A.V. & Jones, B.L. (2006). Extreme behavior of multivariate phase-type distributions. Insurance: Mathematics and Economics, 41(2), 223–233.Google Scholar
Asmussen, S. (2003). Applied Probabilities and Queues. Springer, New York.Google Scholar
Asmussen, S. & Albrecher, H. (2010). Ruin Probabilities, 2nd edition. World Scientific Publishing, Singapore.Google Scholar
Asmussen, S., Jensen, J.L. & Rojas-Nandayapa, L. (2016). Exponential family techniques for the log-normal left tail. Scandinavian Journal of Statistics, 43(3), 774–787.Google Scholar
Asmussen, S., Nerman, O. & Olsson, M. (1996). Fitting phase-type distributions via the EM algorithm. Scandinavian Journal of Statistics, 23, 419–441.Google Scholar
Assaf, D. & Levikson, B. (1982). Closure of phase type distributions under operations arising in reliability theory. The Annals of Probability, 10(1), 265–269.Google Scholar
Bingham, N.H., Goldie, C.M. & Teugels, J.L. (1987). Regular Variation. Cambridge University Press, Cambridge.Google Scholar
Bladt, M., Campillo Navarro, A. & Nielsen, B. (2015a). On the use of functional calculus for phase-type and related distributions. Stochastic Models, 32(1), 1–19.Google Scholar
Bladt, M., Nielsen, B.F. & Samorodnitsky, G. (2015b). Calculation of ruin probabilities for a dense class of heavy-tailed distributions. Scandinavian Actuarial Journal, 2015(7), 573–591.Google Scholar
Bladt, M. & Rojas-Nandayapa, L. (2017). Fitting phase-type scale mixtures to heavy-tailed data and distributions. Submitted arXiv:1705.04357.Google Scholar
Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory of Probability and its Applications, 10(2), 323–331.Google Scholar
Damen, E., Mikosch, T., Rosinski, J. & Samorodnitsky, G. (2014). General inverse problems for regular variation. Journal of Applied Probability, 51A, 229–248.Google Scholar
de Haan, L. (1970). On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Mathematical Centre tracts. Mathematisch Centrum, Amsterdam.Google Scholar
Denisov, D. & Zwart, B. (2007). On a theorem of Breiman and a class of random difference equations. Journal of Applied Probability, 44(4), 1031–1046.Google Scholar
Embrechts, P. & Goldie, C.M. (1980). On closure and factorization properties of subexponential and related distributions. Journal of the Australian Mathematical Society (Series A), 29, 243–256.Google Scholar
Foss, S., Korshunov, D. & Zachary, S. (2011). An Introduction to Heavy-Tailed and Subexponential Distributions. Springer, New York.Google Scholar
Goldie, C.M. & Resnick, S.I. (1988). Distributions that are both subexponential and in the domain of attraction of an extreme value distribution. Advances in Applied Probability, 20(4), 706–718.Google Scholar
Greiner, M., Jobmann, M. & Lipsky, L. (1999). The importance of power-tail distributions for modelling queueing systems. Operations Research, 47(2), 313–326.Google Scholar
Hashorva, E., Pakes, A.G. & Tang, Q. (2010). Asymptotics of random contractions. Insurance: Mathematics and Economics, 47, 405–414.Google Scholar
Jacobsen, M., Mikosch, T. & Rosinski, J. (2009). Inverse problems for regular variation of linear filters, a cancellation property for a-finite measures and identification of stable laws. The Annals of Applied Probability, 19(1), 210–242.Google Scholar
Jessen, A.H. & Mikosch, T. (2006). Regularly varying functions. Publications de L’Institut Mathematique, 79(93), 171–192.Google Scholar
Kang, S. & Serfozo, R.F. (1999). Extreme values of phase-type and mixed random variables with parallel-processing examples. Journal of Applied Probability, 39(1), 194–210.Google Scholar
Latouche, G. & Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. American Statistical and the Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
Liu, Y. & Tang, Q. (2010). The subexponential product convolution of two Weibull-type distributions. Journal of the Australian Mathematical Society, 89, 277–288.Google Scholar
Maier, R.S. & O’Cinneide, C.A. (1992). A closure characterisation of phase-type distributions. Journal of Applied Probability, 29(1), 92–103.Google Scholar
Neuts, M.F. (1981). Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach. John Hopkins University Press, Baltimore, MD.Google Scholar
Peralta, O., Rojas-Nandayapa, L., Xie, W. & Yao, H. (2016). Approximation of ruin probabilities via erlangized scale mixtures. Submitted arXiv:1705.09405.Google Scholar
Ragab, F.M. (1965). Multiple integrals involving product of modified Bessel functions of the second kind. Rendiconti del Circolo Matematico di Palermo, 14(3), 367–381.Google Scholar
Shi, D., Guo, J. & Liu, L. (1996). SPH-distributions and the rectangle-iterative algorithm. In S. Chakravarthy & A.S. Alfa, (Eds.) Matrix-Analytic Methods in Stochastic Models (pp. 207–224). Marcel Dekker, Boca Raton.Google Scholar
Su, C. & Chen, Y. (2006). On the behavior of the product of independent random variables. Science China Series, A49(3), 342–359.Google Scholar
Tang, Q. (2008). From light tails to heavy tails through multiplier. Extremes, 11(4), 379–391.Google Scholar
Vatamidou, E., Adan, I.J.B.F., Vlasiou, M. & Zwart, B. (2013). Corrected phase-type approximations of heavy-tailed risk models using perturbation analysis. Insurance: Mathematics and Economics, 53, 366–378.Google Scholar
Vatamidou, E., Adan, I.J.B.F., Vlasiou, M. & Zwart, B. (2014). On the accuracy of phase-type approximations of heavy-tailed risk models. Scandinavian Actuarial Journal, 2014(6), 510–534.Google Scholar