×

Odd Pareto families of distributions for modeling loss payment data. (English) Zbl 1416.91208

Summary: A three-parameter generalization of the Pareto distribution is presented with density function having a flexible upper tail in modeling loss payment data. This generalized Pareto distribution will be referred to as the odd Pareto distribution since it is derived by considering the distributions of the odds of the Pareto and inverse Pareto distributions. Basic properties of the odd Pareto distribution (OP) are studied. Model parameters are estimated using both modified and regular maximum likelihood methods. Simulation studies are conducted to compare the OP with the exponentiated Pareto, Burr, and Kumaraswamy distributions using two different test statistics based on the ml method. Furthermore, two examples from the Norwegian fire insurance claims data-set are provided to illustrate the upper tail flexibility of the distribution. Extensions of the odd Pareto distribution are also considered to improve the fitting of data.

MSC:

91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
62E15 Exact distribution theory in statistics

Software:

ismev
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Abu Bakar, S. A.; Hamzar, N. A.; Maghsoudi, M.; Nadarajah, S., Modeling loss data using composite models, Insurance: Mathematics and Economics, 61, 146-154, (2015) · Zbl 1314.91130
[2] Akinsete, A. A.; Famoye, F.; Lee, C., The beta-Pareto distribution, Statistics, 42, 6, 547-563, (2008) · Zbl 1274.60033
[3] Aljarrah, M. A.; Famoye, F.; Lee, C., A new Weibull-Pareto distribution, Communications in Statistics – Theory and Methods, 44, 19, 4077-4095, (2015) · Zbl 1338.62051
[4] Arnold, B. C., Pareto distributions, (2015), CRC Press, Boca Raton · Zbl 1361.62004
[5] Beirlant, J.; Joossens, E.; Segers, J., Unbiased tail estimation by an extension of the generalized Pareto distribution. (CentER Discussion Paper; Vol. 2005-112), (2005), Econometrics, Tilburg
[6] Beirlant, J.; Matthys, G.; Dierckx, G., Heavy-tailed distributions and rating, Astin Bulletin, 31, 1, 37-58, (2001)
[7] Beirlant, J.; Teugels, J. L.; Vynckie, P., Practical analysis of extreme values, (1996), Leuven University Press, Leuven
[8] Brazauskas, V.; Kleefeld, A., Folded- and log-folded-t distributions as models for insurance loss data, Scandinavian Acturial Journal, 1, 59-74, (2011) · Zbl 1277.62248
[9] Brazauskas, V.; Kleefeld, A., Authors’ reply to ‘letter to the editor regarding folded models and the paper by brazauskas and kleefeld (2011)’, Scandinavian Acturial Journal, 8, 753-757, (2014) · Zbl 1392.62309
[10] Brazauskas, V.; Kleefeld, A., Modeling severity and measuring tail risk of Norwegian fire claims, North American Actuarial Journal, 20, 1, 1-16, (2016)
[11] Coles, S., An introduction to statistical modeling of extreme values, (2001), Springer, London · Zbl 0980.62043
[12] Cooray, K., Generalization of the Weibull distribution: the odd Weibull family, Statistical Modeling, 6, 265-277, (2006)
[13] Cooray, K., A study of moments and likelihood estimators of the odd Weibull distribution, Statistical Methodology, 26, 72-83, (2015)
[14] Gilchrist, W. G., Statistical modelling with quantile functions, (2000), Chapman & Hall, London
[15] Gupta, R. C.; Gupta, R. D.; Gupta, P. L., Modeling failure time data by lehman alternatives, Communications in Statistics: Theory and Methods, 27, 887-904, (1998) · Zbl 0900.62534
[16] Gupta, R. D.; Kundu, D., Discriminating between Weibull and generalized exponential distributions, Computational Statistics and Data Analysis, 43, 179-196, (2003) · Zbl 1429.62060
[17] Hosking, J. R. M.; Wallis, J. R., Parameter and quantile estimation for the generalized Pareto distribution, Technometrics, 29, 339-349, (1987) · Zbl 0628.62019
[18] Johnson, N. L.; Kotz, S.; Balakrishnan, N., Continuous univariate distributions, (1994), Wiley, New York · Zbl 0811.62001
[19] Kumaraswamy, P., A generalized probability density function for double-bounded random processes, Journal of Hydrology, 46, 1-2, 79-88, (1980)
[20] Nadarajah, S., Exponentiated Pareto distributions, Statistics: A Journal of Theoritical and Applied Statistics, 39, 3, 255-260, (2005) · Zbl 1070.62008
[21] Nadarajah, S.; Bakar, S. A. A., New folded models for the log-transformed Norwegian fire claim data, Communications in Statistics - Theory and Methods, 44, 20, 4408-4440, (2015) · Zbl 1357.62076
[22] Pickands, J., Statistical inference using extreme order statistics, The Annals of Statistics, 3, 119-131, (1975) · Zbl 0312.62038
[23] Scollnik, D. P., Regarding folded models and the paper by brazauskas and kleefeld (2011), Scandinavian Actuarial Journal, 3, 278-281, (2014) · Zbl 1392.62313
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.