Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

A new extended alpha power transformed family of distributions: properties, characterizations and an application to a data set in the insurance sciences

  • Ahmad, Zubair (Department of Statistics, Yazd University) ;
  • Mahmoudi, Eisa (Department of Statistics, Yazd University) ;
  • Hamedani, G.G. (Department of Mathematical and Statistical Sciences, Marquette University)
  • Received : 2019.09.19
  • Accepted : 2020.01.10
  • Published : 2021.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

Heavy tailed distributions are useful for modeling actuarial and financial risk management problems. Actuaries often search for finding distributions that provide the best fit to heavy tailed data sets. In the present work, we introduce a new class of heavy tailed distributions of a special sub-model of the proposed family, called a new extended alpha power transformed Weibull distribution, useful for modeling heavy tailed data sets. Mathematical properties along with certain characterizations of the proposed distribution are presented. Maximum likelihood estimates of the model parameters are obtained. A simulation study is provided to evaluate the performance of the maximum likelihood estimators. Actuarial measures such as Value at Risk and Tail Value at Risk are also calculated. Further, a simulation study based on the actuarial measures is done. Finally, an application of the proposed model to a heavy tailed data set is presented. The proposed distribution is compared with some well-known (i) two-parameter models, (ii) three-parameter models and (iii) four-parameter models.

Keywords

References

  1. Adcock C, Eling M, and Loperfido N (2015). Skewed distributions in finance and actuarial science: a review, The European Journal of Finance, 21, 1253-1281. https://doi.org/10.1080/1351847X.2012.720269
  2. Ahmad Z, Hamedani GG, and Butt NS (2019a). Recent developments in distribution theory: a brief survey and some new generalized classes of distributions, Pakistan Journal of Statistics and Operation Research, 15, 87-110.
  3. Ahmad Z, Ilyas M, and Hamedani GG (2019b). The extended alpha power transformed family of distributions: properties and applications, Journal of Data Science, 17, 726-741.
  4. Artzner P (1999). Application of coherent risk measures to capital requirements in insurance, North American Actuarial Journal, 3, 11-25. https://doi.org/10.1080/10920277.1999.10595795
  5. Bagnato L and Punzo A (2013). Finite mixtures of unimodal beta and gamma densities and the k-bumps algorithm, Computational Statistics, 28, 1571-1597. https://doi.org/10.1007/s00180-012-0367-4
  6. Bakar SA, Hamzah NA, Maghsoudi M, and Nadarajah S (2015). Modeling loss data using composite models, Insurance: Mathematics and Economics, 61, 146-154. https://doi.org/10.1016/j.insmatheco.2014.08.008
  7. Bernardi M, Maruotti A, and Petrella L (2012). Skew mixture models for loss distributions: a Bayesian approach, Insurance: Mathematics and Economics, 51, 617-623. https://doi.org/10.1016/j.insmatheco.2012.08.002
  8. Bhati D and Ravi S (2018). On generalized log-Moyal distribution: A new heavy tailed size distribution, Insurance: Mathematics and Economics, 79, 247-259. https://doi.org/10.1016/j.insmatheco.2018.02.002
  9. Cooray K and Ananda MM (2005). Modeling actuarial data with a composite lognormal-Pareto model, Scandinavian Actuarial Journal, 2005, 321-334. https://doi.org/10.1080/03461230510009763
  10. Eling M (2012). Fitting insurance claims to skewed distributions: Are the skew-normal and skewstudent good models?, Insurance: Mathematics and Economics, 51, 239-248. https://doi.org/10.1016/j.insmatheco.2012.04.001
  11. Garcia VJ, Gomez-Deniz E, and Vazquez-Polo FJ (2014). On modelling insurance data by using a generalized lognormal distribution, Revista de Metodos Cuantitativos para la Economia y la Empresa, 18, 146-162.
  12. Glanzel W (1987). A characterization theorem based on truncated moments and its application to some distribution families. In Mathematical Statistics and Probability Theory (pp. 75-84), Springer, Dordrecht.
  13. Glanzel W (1990). Some consequences of a characterization theorem based on truncated moments, Statistics, 21, 613-618. https://doi.org/10.1080/02331889008802273
  14. Ibragimov R and Prokhorov A (2017). Heavy tails and copulas: topics in dependence modelling in economics and finance.
  15. Kazemi R and Noorizadeh M (2015). A comparison between skew-logistic and skew-normal distributions, MATEMATIKA: Malaysian Journal of Industrial and Applied Mathematics, 31, 15-24.
  16. Klugman SA, Panjer HH, and Willmot GE (2012). Loss Models: From Data to Decisions (4th ed), John Wiley and Sons, Hoboken, NJ.
  17. Landsman Z, Makov U, and Shushi T (2016). Tail conditional moments for elliptical and log-elliptical distributions, Insurance: Mathematics and Economics, 71, 179-188. https://doi.org/10.1016/j.insmatheco.2016.09.001
  18. Lane MN (2000). Pricing risk transfer transactions 1, ASTIN Bulletin: The Journal of the IAA, 30, 259-293. https://doi.org/10.2143/AST.30.2.504635
  19. Mahdavi A and Kundu D (2017). A new method for generating distributions with an application to exponential distribution, Communications in Statistics-Theory and Methods, 46, 6543-6557. https://doi.org/10.1080/03610926.2015.1130839
  20. Punzo A, Bagnato L, and Maruotti A (2018). Compound unimodal distributions for insurance losses, Insurance: Mathematics and Economics, 81, 95-107. https://doi.org/10.1016/j.insmatheco.2017.10.007
  21. Reynkens T, Verbelen R, Beirlant J, and Antonio K (2017). Modelling censored losses using splicing: A global fit strategy with mixed Erlang and extreme value distributions, Insurance: Mathematics and Economics, 77, 65-77. https://doi.org/10.1016/j.insmatheco.2017.08.005