From light tails to heavy tails through multiplier

Let X 1 and X 2 be two independent and nonnegative random variables with distributions F 1 and F 2 , respectively. This paper proves that if both F 1 and F 2 are of Weibull type and fulfill certain easily verifiable conditions, then the distribution of the product X 1 X 2 , called the product convolution of F 1 and F 2 , belongs to the class S * and, hence, is subexponential.2000 Mathematics subject classification: primary 60E05; secondary 62E20.

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“…Hence, it is plausible that if Π G is close enough to the integrated tail distribution F of the claim sizes, then we can use ψ Π G (u) as an approximation for ψ F (u), the ruin probability of a Cramér-Lundberg process having claim size distribution F . One of the key features of the class of phase-type scale mixtures is that if Π has unbounded support, then Π G is a heavy-tailed distribution (Rojas-Nandayapa and Xie, 2017; Su and Chen, 2006;Tang, 2008), confirming the hypothesis that the class of phase-type scale mixtures is more appropriate for approximating tail-dependent quantities involving heavytailed distributions. In contrast, the class of classical phase-type distributions is light-tailed and approximations derived from this approach may be inaccurate in the tails (see also Vatamidou et al, 2014, for an extended discussion).…”

Section: Introductionmentioning

confidence: 91%

“…For instance, Su and Chen (2006) show that if two random variables S 1 and S 2 are such that the distribution of S 1 is in L(λ) with λ > 0 and S 2 has unbounded support, then the distribution of S 1 · S 2 is in L(0) (long-tailed), and thus heavy-tailed (see also Tang, 2008). If one further assumes that S 2 is Weibullian with parameter 0 < p 1, then Liu and Tang (2010) show that the product S 1 · S 2 is subexponential.…”

Section: Asymptotic Tail Behaviormentioning

confidence: 99%

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Modelling heavy-tails with phase-type scale mixture distributions

Xie¹

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In this thesis, we present the new results found in exploring the asymptotic tail probability of the phase-type scale mixture distributions, and apply a subclass of the phase-type scale mixtures, the class of Erlangized scale mixture distributions, to the approximation of the probability of ruin.We consider the class of phase-type scale mixture distributions, which can be written as MellinStieltjes convolution Π G of a phase-type distribution G and a nonnegative but otherwise arbitrary distribution Π. Such a class can also be seen as the class of distributions of the product of two independent random variables: a phase-type random variable Y ∼ G and a nonnegative random variable S ∼ Π. We call S the scaling random variable and its corresponding distribution Π the scaling distribution. 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.In this thesis, particular focus is on phase-type scale mixture distributions where the scaling distribution has discrete support -such a class has been recently used in risk applications to approximate heavy-tailed distributions. We extend an existing scheme for numerically calculating the probability of ruin of a classical Cramér-Lundberg reserve process having absolutely continuous but otherwise general claim size distributions. We employ a subclass of the phasetype scale mixtures, which is also a dense class of distributions that we denominate Erlangized scale mixtures (ESM). Such a class corresponds to nonnegative and absolutely continuous distributions which can be written as a Mellin-Stieltjes convolution Π G m of a discrete nonnegative distribution Π with an Erlang distribution G m . A distinctive feature of such a class is that it contains heavy-tailed distributions when the scaling distributions have unbounded support, and it provides sharp approximations to heavy-tailed distributions.We suggest a simple methodology for constructing a sequence of distributions having the form Π G m with the purpose of approximating the integrated tail distribution of the claim sizes.Then we adapt a recent result which delivers an explicit expression for the probability of ruin in the case that the claim size distribution is modelled as an Erlangized scale mixture. We provide simplified expressions for the approximation of the probability of ruin and construct explicit bounds for the error of approximation. We complement our results with a classical example where the claim sizes are heavy-tailed.ii

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From light tails to heavy tails through multiplier

Cited by 51 publications

References 21 publications

Some properties of the exponential distribution class with applications to risk theory

Some properties of the exponential distribution class with applications to risk theory

The Subexponential Product Convolution of Two Weibull-Type Distributions

Modelling heavy-tails with phase-type scale mixture distributions

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