Conditional tail expectations for multivariate phase-type distributions. (English) Zbl 1079.62022

Cai, Jun; Li, Haijun

J. Appl. Probab. 42, No. 3, 810-825 (2005).

Summary: The conditional tail expectation in risk analysis describes the expected amount of risk that can be experienced given that a potential risk exceeds a threshold value, and provides an important measure of right-tail risk. We study the convolution and extreme values of dependent risks that follow a multivariate phase-type distribution, and derive explicit formulae for several conditional tail expectations of the convolution and extreme values for such dependent risks. Utilizing the underlying Markovian property of these distributions, our method not only provides structural insight, but also yields some new distributional properties of multivariate phase-type distributions.

MSC:

62E15	Exact distribution theory in statistics
62P05	Applications of statistics to actuarial sciences and financial mathematics
91B30	Risk theory, insurance (MSC2010)
60J20	Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
62N05	Reliability and life testing

Keywords:

order statistics; univariate phase-type distribution; multivariate phase-type distribution; continuous-time Markov chain; convolution; extreme value; Marshall-Olkin distribution; conditional tail expectation; excess loss; residual lifetime

References:

[1]	Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risks. Math . Finance 9 , 203–228. · Zbl 0980.91042
[2]	Asmussen, S. (2000). Ruin Probabilities . World Scientific, Singapore. · Zbl 0960.60003
[3]	Asmussen, S. (2003). Applied Probability and Queues , 2nd edn. Springer, New York. · Zbl 1029.60001
[4]	Assaf, D., Langberg, N., Savits, T. H. and Shaked, M. (1984). Multivariate phase-type distributions. Operat . Res. 32 , 688–702. · Zbl 0558.60070
[5]	Bowers, N. L. et al . (1997). Actuarial Mathematics . The Society of Actuaries, Schaumburg, IL.
[6]	Cai, J. and Li, H. (2005). Multivariate risk model of phase type. Insurance Math . Econom. 36 , 137–152. · Zbl 1122.60049
[7]	Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance . Springer, Berlin. · Zbl 0873.62116
[8]	Kulkarni, V. G. (1989). A new class of multivariate phase type distributions. Operat . Res. 37 , 151–158. · Zbl 0667.60019
[9]	Landsman, Z. and Valdez, E. (2003). Tail conditional expectations for elliptical distributions. N . Amer. Actuarial J. 7 , 55–71. · Zbl 1084.62512
[10]	Li, H. and Xu, S. H. (2000). On the dependence structure and bounds of correlated parallel queues and their applications to synchronized stochastic systems. J . Appl. Prob. 37 , 1020–1043. · Zbl 0981.60087
[11]	Marshall, A. W. and Olkin, I. (1967). A multivariate exponential distribution. J . Amer. Statist. Assoc. 2 , 84–98. · Zbl 0147.38106
[12]	Neuts, M. F. (1981). Matrix -Geometric Solutions in Stochastic Models. An Algorithmic Approach. Johns Hopkins University Press, Baltimore, MD. · Zbl 0469.60002
[13]	Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Finance and Insurance . John Wiley, New York. · Zbl 0940.60005
[14]	Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications . Academic Press, New York. · Zbl 0806.62009