Bladt, Mogens3; Nielsen, Bo Friis1; Samorodnitsky, Gennady5
1 Department of Applied Mathematics and Computer Science, Technical University of Denmark2 Statistics and Data Analysis, Department of Applied Mathematics and Computer Science, Technical University of Denmark3 National University of Mexico4 Cornell University5 Cornell University
In this paper, we propose a class of infinite-dimensional phase-type distributions with finitely many parameters as models for heavy tailed distributions. The class of finite-dimensional phase-type distributions is dense in the class of distributions on the positive reals and may hence approximate any such distribution. We prove that formulas from renewal theory, and with a particular attention to ruin probabilities, which are true for common phase-type distributions also hold true for the infinite-dimensional case. We provide algorithms for calculating functionals of interest such as the renewal density and the ruin probability. It might be of interest to approximate a given heavy tailed distribution of some other type by a distribution from the class of infinite-dimensional phase-type distributions and to this end we provide a calibration procedure which works for the approximation of distributions with a slowly varying tail. An example from risk theory, comparing ruin probabilities for a classical risk process with Pareto distributed claim sizes, is presented and exact known ruin probabilities for the Pareto case are compared to the ones obtained by approximating by an infinite-dimensional hyper-exponential distribution.
Scandinavian Actuarial Journal, 2015, Vol 2015, Issue 7, p. 573-591
Heavy tails; Phase-type distributions; Ruin probability