Corrected phase-type approximations of heavy-tailed risk models using perturbation analysis

Vatamidou, Eleni; Adan, Ijbf Ivo; Vlasiou, Maria; Zwart, AP Bert

doi:10.1016/j.insmatheco.2013.07.002

Cited by 13 publications

(19 citation statements)

References 34 publications

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“…The Laplace transform of the latter distribution is F h (s ) = 1 − For this combination of service time distributions, the survival queueing delay can be found explicitly, by following the same ideas as in Theorem 9 of Ref. [36] .…”

Section: Numerical Experimentsmentioning

confidence: 99%

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Corrected Phase-Type Approximations of Heavy-Tailed Queueing Models in a Markovian Environment

et al. 2014

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2 We develop accurate approximations for the delay distribution of the MArP/G/1 queue that capture the exact tail behavior and provide bounded relative errors. Motivated by statistical analysis, we consider the service times as a mixture of a phase-type and a heavy-tailed distribution. With the aid of perturbation analysis, we derive corrected phase-type approximations as a sum of the delay in a MArP/PH/1 queue and a heavy-tailed component depending on the perturbation parameter. We exhibit their performance with numerical examples.

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Section: Numerical Experimentsmentioning

confidence: 99%

“…Therefore, we call this term correction term, and inspired by the terminology corrected heavy traffic approximations [7] we refer to our approximations as corrected phase-type approximations. In a previous study [36] , we applied this approach to Poisson arrivals.…”

Section: Introductionmentioning

confidence: 99%

Corrected Phase-Type Approximations of Heavy-Tailed Queueing Models in a Markovian Environment

et al. 2014

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“…Similar attempts to approximate the probability of ruin for the Cramér-Lundberg model have been made before (see Santana et al, 2017;Vatamidou et al, 2013). However, our contribution is that we propose to directly approximate the integrated tail distribution in approximation A instead of the claim size distribution.…”

Section: Introductionmentioning

confidence: 90%

“…Furthermore, Bladt and Rojas-Nandayapa (2017) provide statistical inference procedures based on the EM algorithm to adjust phase-type scale mixtures to heavy-tailed data/distributions. Other references of interest that apply similar ideas to risk models include Hashorva et al (2010) and Vatamidou et al (2013).…”

Section: Introductionmentioning

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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Efficient Simulation of Ruin Probabilities When Claims are Mixtures of Heavy and Light Tails

Albrecher

Bladt

Vatamidou

2020

Methodol Comput Appl Probab

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Corrected phase-type approximations of heavy-tailed risk models using perturbation analysis

Cited by 13 publications

References 34 publications

Corrected Phase-Type Approximations of Heavy-Tailed Queueing Models in a Markovian Environment

Corrected Phase-Type Approximations of Heavy-Tailed Queueing Models in a Markovian Environment

Modelling heavy-tails with phase-type scale mixture distributions

Efficient Simulation of Ruin Probabilities When Claims are Mixtures of Heavy and Light Tails

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