14,270 research outputs found
On the accuracy of phase-type approximations of heavy-tailed risk models
Numerical evaluation of ruin probabilities in the classical risk model is an
important problem. If claim sizes are heavy-tailed, then such evaluations are
challenging. To overcome this, an attractive way is to approximate the claim
sizes with a phase-type distribution. What is not clear though is how many
phases are enough in order to achieve a specific accuracy in the approximation
of the ruin probability. The goals of this paper are to investigate the number
of phases required so that we can achieve a pre-specified accuracy for the ruin
probability and to provide error bounds. Also, in the special case of a
completely monotone claim size distribution we develop an algorithm to estimate
the ruin probability by approximating the excess claim size distribution with a
hyperexponential one. Finally, we compare our approximation with the heavy
traffic and heavy tail approximations.Comment: 24 pages, 13 figures, 8 tables, 38 reference
Asymptotic tail behavior of phase-type scale mixture distributions
We consider phase-type scale mixture distributions which correspond to
distributions of a product of two independent random variables: a phase-type
random variable and a nonnegative but otherwise arbitrary random variable
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 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.Comment: 18 pages, 0 figur
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