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

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Abstract

Numerical evaluation of performance measures in heavy-tailed risk models is an important and challenging problem. In this paper, we construct very accurate approximations of such performance measures that provide small absolute and relative errors. Motivated by statistical analysis, we assume that the claim sizes are a mixture of a phase-type and a heavy-tailed distribution and with the aid of perturbation analysis we derive a series expansion for the performance measure under consideration. Our proposed approximations consist of the first two terms of this series expansion, where the first term is a phase-type approximation of our measure. We refer to our approximations collectively as corrected phase-type approximations. We show that the corrected phase-type approximations exhibit a nice behavior both in finite and infinite time horizon, and we check their accuracy through numerical experiments. Keywords: Ruin probability; Heavy-tailed claim sizes; Error bounds; Tail asymptotics; Relative errors; Value at risk
Original languageEnglish
Pages (from-to)366-378
JournalInsurance: Mathematics and Economics
Volume53
Issue number2
DOIs
Publication statusPublished - 2013

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Perturbation Analysis
Model Analysis
Approximation
Performance Measures
Relative Error
Series Expansion
Tail Asymptotics
Heavy-tailed Distribution
Ruin Probability
Value at Risk
Term
Error Bounds
Statistical Analysis
Perturbation
Risk model
Horizon
Numerical Experiment
Evaluation
Performance measures

Cite this

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title = "Corrected phase-type approximations of heavy-tailed risk models using perturbation analysis",
abstract = "Numerical evaluation of performance measures in heavy-tailed risk models is an important and challenging problem. In this paper, we construct very accurate approximations of such performance measures that provide small absolute and relative errors. Motivated by statistical analysis, we assume that the claim sizes are a mixture of a phase-type and a heavy-tailed distribution and with the aid of perturbation analysis we derive a series expansion for the performance measure under consideration. Our proposed approximations consist of the first two terms of this series expansion, where the first term is a phase-type approximation of our measure. We refer to our approximations collectively as corrected phase-type approximations. We show that the corrected phase-type approximations exhibit a nice behavior both in finite and infinite time horizon, and we check their accuracy through numerical experiments. Keywords: Ruin probability; Heavy-tailed claim sizes; Error bounds; Tail asymptotics; Relative errors; Value at risk",
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Corrected phase-type approximations of heavy-tailed risk models using perturbation analysis. / Vatamidou, E.; Adan, I.J.B.F.; Vlasiou, M.; Zwart, B.

In: Insurance: Mathematics and Economics, Vol. 53, No. 2, 2013, p. 366-378.

Research output: Contribution to journalArticleAcademicpeer-review

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AB - Numerical evaluation of performance measures in heavy-tailed risk models is an important and challenging problem. In this paper, we construct very accurate approximations of such performance measures that provide small absolute and relative errors. Motivated by statistical analysis, we assume that the claim sizes are a mixture of a phase-type and a heavy-tailed distribution and with the aid of perturbation analysis we derive a series expansion for the performance measure under consideration. Our proposed approximations consist of the first two terms of this series expansion, where the first term is a phase-type approximation of our measure. We refer to our approximations collectively as corrected phase-type approximations. We show that the corrected phase-type approximations exhibit a nice behavior both in finite and infinite time horizon, and we check their accuracy through numerical experiments. Keywords: Ruin probability; Heavy-tailed claim sizes; Error bounds; Tail asymptotics; Relative errors; Value at risk

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