The Heavy-Tailed Exponential Distribution: Risk Measures, Estimation, and Application to Actuarial Data
Modeling insurance data using heavy-tailed distributions is of great interest for actuaries. Probability distributions present a description of risk exposure, where the level of exposure to the risk can be determined by “key risk indicators” that usually are functions of the model. Actuaries and ris...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/8/1276 |
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| author | Ahmed Z. Afify Ahmed M. Gemeay Noor Akma Ibrahim |
| author_facet | Ahmed Z. Afify Ahmed M. Gemeay Noor Akma Ibrahim |
| author_sort | Ahmed Z. Afify |
| collection | DOAJ |
| description | Modeling insurance data using heavy-tailed distributions is of great interest for actuaries. Probability distributions present a description of risk exposure, where the level of exposure to the risk can be determined by “key risk indicators” that usually are functions of the model. Actuaries and risk managers often use such key risk indicators to determine the degree to which their companies are subject to particular aspects of risk, which arise from changes in underlying variables such as prices of equity, interest rates, or exchange rates. The present study proposes a new heavy-tailed exponential distribution that accommodates bathtub, upside-down bathtub, decreasing, decreasing-constant, and increasing hazard rates. Actuarial measures including value at risk, tail value at risk, tail variance, and tail variance premium are derived. A computational study for these actuarial measures is conducted, proving that the proposed distribution has a heavier tail as compared with the alpha power exponential, exponentiated exponential, and exponential distributions. We adopt six estimation approaches for estimating its parameters, and assess the performance of these estimators via Monte Carlo simulations. Finally, an actuarial real data set is analyzed, proving that the proposed model can be used effectively to model insurance data as compared with fifteen competing distributions. |
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| format | Article |
| id | doaj.art-d9e5826357f94105b12e8d18b0e71a11 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T18:00:25Z |
| publishDate | 2020-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-d9e5826357f94105b12e8d18b0e71a112023-11-20T08:56:40ZengMDPI AGMathematics2227-73902020-08-0188127610.3390/math8081276The Heavy-Tailed Exponential Distribution: Risk Measures, Estimation, and Application to Actuarial DataAhmed Z. Afify0Ahmed M. Gemeay1Noor Akma Ibrahim2Department of Statistics, Mathematics and Insurance, Benha University, Benha 13511, EgyptDepartment of Mathematics, Faculty of Science, Tanta University, Tanta 31527, EgyptInstitute for Mathematical Research, Universiti Putra Malaysia, Selangor 43400, MalaysiaModeling insurance data using heavy-tailed distributions is of great interest for actuaries. Probability distributions present a description of risk exposure, where the level of exposure to the risk can be determined by “key risk indicators” that usually are functions of the model. Actuaries and risk managers often use such key risk indicators to determine the degree to which their companies are subject to particular aspects of risk, which arise from changes in underlying variables such as prices of equity, interest rates, or exchange rates. The present study proposes a new heavy-tailed exponential distribution that accommodates bathtub, upside-down bathtub, decreasing, decreasing-constant, and increasing hazard rates. Actuarial measures including value at risk, tail value at risk, tail variance, and tail variance premium are derived. A computational study for these actuarial measures is conducted, proving that the proposed distribution has a heavier tail as compared with the alpha power exponential, exponentiated exponential, and exponential distributions. We adopt six estimation approaches for estimating its parameters, and assess the performance of these estimators via Monte Carlo simulations. Finally, an actuarial real data set is analyzed, proving that the proposed model can be used effectively to model insurance data as compared with fifteen competing distributions.https://www.mdpi.com/2227-7390/8/8/1276risk measuresexponential distributionparameter estimationtail variance premiumVaRTVaR |
| spellingShingle | Ahmed Z. Afify Ahmed M. Gemeay Noor Akma Ibrahim The Heavy-Tailed Exponential Distribution: Risk Measures, Estimation, and Application to Actuarial Data Mathematics risk measures exponential distribution parameter estimation tail variance premium VaR TVaR |
| title | The Heavy-Tailed Exponential Distribution: Risk Measures, Estimation, and Application to Actuarial Data |
| title_full | The Heavy-Tailed Exponential Distribution: Risk Measures, Estimation, and Application to Actuarial Data |
| title_fullStr | The Heavy-Tailed Exponential Distribution: Risk Measures, Estimation, and Application to Actuarial Data |
| title_full_unstemmed | The Heavy-Tailed Exponential Distribution: Risk Measures, Estimation, and Application to Actuarial Data |
| title_short | The Heavy-Tailed Exponential Distribution: Risk Measures, Estimation, and Application to Actuarial Data |
| title_sort | heavy tailed exponential distribution risk measures estimation and application to actuarial data |
| topic | risk measures exponential distribution parameter estimation tail variance premium VaR TVaR |
| url | https://www.mdpi.com/2227-7390/8/8/1276 |
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