On the accuracy of phase-type approximations of heavy-tailed risk models

Vatamidou, Eleni; Adan, I.J.B.F.; Vlasiou, Maria; Zwart, A.P.

doi:10.1080/03461238.2012.729154

Cited by 20 publications

(30 citation statements)

References 34 publications

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“…The following theorem provides a refinement for the upper bound provided in Vatamidou et al (2014). As mentioned before, this result will be used later in the construction of upper bounds for each individual error.…”

Section: General Bounds For the Error Of Approximationmentioning

confidence: 99%

“…In Subsection 4.4.1, we refine an existing upper bound introduced in Vatamidou et al (2014) for the distance between two ruin probabilities…”

Section: Error Bounds For the Ruin Probabilitymentioning

confidence: 99%

“…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: 99%

“…Therefore, the PH class cannot correctly capture the characteristic behavior of a heavy-tailed distribution in spite of its denseness. In fact, this approach may deliver unreliable approximations for important quantities of interest, such as the ruin probability of a Cramér-Lundberg risk process with heavy-tailed claim size distributions (Vatamidou et al, 2014). As an alternative, the PH class has been extended to distributions of absorption times having a countable number of transient states (this approach is attributed to Neuts, 1981).…”

Section: Introductionmentioning

confidence: 99%

“…However, the phase-type distributions are inherently light-tailed, they cannot capture the heavy-tailed behaviors precisely (cf. Vatamidou et al, 2014).…”

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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Section: General Bounds For the Error Of Approximationmentioning

confidence: 99%

“…In Subsection 4.4.1, we refine an existing upper bound introduced in Vatamidou et al (2014) for the distance between two ruin probabilities…”

Section: Error Bounds For the Ruin Probabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…However, the phase-type distributions are inherently light-tailed, they cannot capture the heavy-tailed behaviors precisely (cf. Vatamidou et al, 2014).…”

Section: Introductionmentioning

confidence: 99%

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Modelling heavy-tails with phase-type scale mixture distributions

Xie¹

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Phase Transitions in Multiserver Queuing Systems

Tsitsiashvili¹,

Osipova

2016

Communications in Computer and Information Science

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Evaluating readmission rates and discharge planning by analyzing the length-of-stay of patients

Fan

Liao

2018

Ann Oper Res

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The length-of-stay (LOS) is an important quality metric in health care, and the use of phase-type (PH) distribution provides a flexible method for modeling LOS. In this paper, we model the patient flow information collected in a hospital for patients of distinct diseases, including headache, liveborn infant, alcohol abuse, acute upper respiratory infection, and secondary cataract. Based on the results obtained from fitting Coxian PH distributions to the LOS data, the patients can be divided into different groups. By analyzing each group to find out their common characteristics, the corresponding readmission rate and other useful information can be evaluated. Furthermore, a comparison of patterns for each disease is analyzed. We conclude that it is important to offering better service and avoiding waste of sources, by the analysis of the relations between groups and readmission. In addition, comparing the patterns within distinct diseases, a better decision for assigning resources and improving the insurance policy can be made.

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On the accuracy of phase-type approximations of heavy-tailed risk models

Cited by 20 publications

References 34 publications

Modelling heavy-tails with phase-type scale mixture distributions

Modelling heavy-tails with phase-type scale mixture distributions

Phase Transitions in Multiserver Queuing Systems

Evaluating readmission rates and discharge planning by analyzing the length-of-stay of patients

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