[Paper Review] Asymptotic tail behavior of phase-type scale mixture distributions

Leonardo Rojas‐Nandayapa, Wangyue Xie|arXiv (Cornell University)|Feb 6, 2015

Probability and Risk Models27 references3 citations

TL;DR

This paper investigates the asymptotic tail behavior of phase-type scale mixture distributions—defined as products of a phase-type random variable and a nonnegative scaling random variable—providing conditions for light- or heavy-tailed behavior, subexponentiality, and maximum domains of attraction. It establishes that discrete scaling distributions enable tractable approximation of heavy-tailed distributions, with tail behavior closely tied to the scaling distribution's support and tail properties.

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 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.

Motivation & Objective

• To understand the tail behavior of phase-type scale mixture distributions, particularly in relation to the scaling distribution's properties.
• To determine when such mixtures are light- or heavy-tailed, and to classify their maximum domains of attraction.
• To provide theoretical justification for using discrete scaling distributions in approximating heavy-tailed distributions in risk modeling.
• To establish asymptotic approximations for the tail of the mixture distribution based on the scaling distribution's tail.
• To support practical applications in insurance and risk theory by linking distributional properties to modeling choices in phase-type scale mixtures.

Proposed method

• Uses the Mellin–Stieltjes convolution to define the phase-type scale mixture distribution as $ F(x) = \int_0^\infty G(x/s) \, dH(s) $, where $ G $ is phase-type and $ H $ is the scaling distribution.
• Applies Breiman’s lemma to derive asymptotic approximations for the tail of the mixture distribution, particularly when $ H $ has a regularly varying tail.
• Analyzes the case of discrete scaling distributions by modeling the support as a geometric progression to ensure fast-converging infinite series.
• Derives asymptotic bounds and approximations for the survival function $ \overline{F}(x) $, showing proportionality to $ \overline{H}(x) $ under certain moment conditions.
• Uses the Laplace transform and matrix-exponential representations to analyze the tail behavior of phase-type components.
• Establishes conditions under which the mixture inherits subexponentiality from the scaling distribution, particularly when $ H $ is subexponential and $ G $ has finite moments.

Experimental results

Research questions

• RQ1 Under what conditions on the scaling distribution $ H $ is the phase-type scale mixture distribution heavy-tailed?
• RQ2 How does the choice of discrete support for $ H $, particularly geometric progression, affect the tail approximation of the mixture?
• RQ3 What are the maximum domains of attraction for phase-type scale mixture distributions, and how do they relate to the scaling distribution?
• RQ4 Can phase-type scale mixtures with discrete $ H $ achieve asymptotic tail behavior equivalent to regularly varying or Weibullian distributions?
• RQ5 What are the theoretical foundations for using such mixtures in approximating heavy-tailed claim size distributions in ruin probability models?

Key findings

• Phase-type scale mixture distributions are heavy-tailed if the scaling distribution $ H $ has unbounded support and regularly varying tails.
• When the scaling distribution $ H $ is regularly varying with index $ -\alpha $, the mixture tail $ \overline{F}(x) $ is asymptotically proportional to $ \overline{H}(x) $, with a constant factor involving the moment-generating function of $ G $.
• For a discrete scaling distribution with geometric support, the infinite series representation converges rapidly, enabling efficient numerical computation and truncation in practice.
• The asymptotic approximation $ \overline{F}(x) \approx \frac{1 - e^{-\alpha/K}}{\alpha/K} M_G(\alpha) \overline{H}(x) $ is consistent with theoretical bounds and holds under moment conditions on $ G $.
• Subexponentiality of the mixture is inherited from $ H $ when $ G $ has finite moments and $ H $ is subexponential, ensuring the tail is dominated by the maximum of i.i.d. components.
• The method of geometric discretization of $ H $ allows for accurate and computationally efficient approximation of heavy-tailed distributions such as Pareto in ruin probability models.