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 nonnegative 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.Comment: 18 pages, 0 figur

Rojas-Nandayapa, Leonardo

Xie, Wangyue

English

arXiv

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 nonnegative 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

University of Liverpool Repository

Asymptotic tail behavior of phase-type scale mixture distributions

We consider phase–type scale mixture distributions which correspond to distri-butions of random variables obtained as the product of two random variables: a phase–type random variable Y and a nonnegative but otherwise arbitrary random variable S called the scaling random variable. We prove that the distri-bution of X: = S · Y is heavy–tailed iff the distribution of the scaling random variable S has unbounded support. In addition, we prove that phase–type scale mixtures preserve the maximum domain of attraction of the distribution of the scaling random variable S. For the Fréchet case, we prove a converse of Breiman’s lemma which provides the exact asymptotics of the tail distribu-tion of X. Finally, we explore subexponentiality for some important cases of phase–type scale mixtures. We complement our results with several examples

Asymptotic tail behavior of phase-type scale mixture distributions

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