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MODELLING INSURANCE DATA WITH THE PARETO ARCTAN DISTRIBUTION

Published online by Cambridge University Press:  19 June 2015

Emilio Gómez-Déniz  and
Enrique Calderín-Ojeda
Emilio Gómez-Déniz*
Affiliation:
Department of Quantitative Methods in Economics and TiDES Institute, University of Las Palmas de Gran Canaria, Spain
Enrique Calderín-Ojeda
Affiliation:
Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Australia E-Mail: ecalderin@unimelb.edu.au
*
E-Mail: emilio.gomez-deniz@ulpgc.es
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Abstract

In this paper, a new methodology based on the use of the inverse of the circular tangent function that allows us to add a scale parameter (say α) to an initial survival function is presented. The latter survival function is determined as limiting case when α tends to zero. By choosing as parent the classical Pareto survival function, the Pareto ArcTan (PAT) distribution is obtained. After providing a comprehensive analysis of its statistical properties, theoretical results with reference to insurance are illustrated. Its performance is compared, by means of the well-known Norwegian fire insurance data, with other existing heavy-tailed distributions in the literature such as Pareto, Stoppa, Shifted Lognormal, Inverse Gamma and Fréchet distributions.

Keywords

Limited expected valuelossPareto distributionshifted lognormal distributionStoppa sistributionthreshold
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 45 , Issue 3 , September 2015 , pp. 639 - 660
Copyright
Copyright © Astin Bulletin 2015 

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References

Arnold, B.C. (1983) Pareto Distributions. Silver Spring, MD: International Cooperative Publishing House.Google Scholar
Beirlant, J., Teugels, J.L and Vynckier, P. (1996) Practical Analysis of Extreme Values. Leuven, Belgium: Leuven University Press.Google Scholar
Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987) Encyclopedia of Mathematics and its Applications (No. 27) Regular Variation. Cambridge: Cambridge University Press.Google Scholar
Boyd, A.V. (1988) Fitting the truncated Pareto distribution to loss distributions. Journal of the Staple Inn Actuarial Society, 31, 151–158.CrossRefGoogle Scholar
Bozdogan, H. (1987) Model selection and Akaike's Information Criterion (AIC): The general theory and its analytical extensions. Psychometrika, 52 (3), 345–370.CrossRefGoogle Scholar
Brazauskas, V. and Kleefeld, A. (2011) Folded and log–folded–t distributions as models for insurance loss data. Scandinavian Actuarial Journal, 2011 (1), 59–74.CrossRefGoogle Scholar
Brazauskas, V. and Kleefeld, A. (2014) Authors' reply to “Letter to the Editor: Regarding folded models and the paper by Brazauskas and Kleefeld (2011)" by Scollnik. Scandinavian Actuarial Journal, 2014 (8), 753–757.CrossRefGoogle Scholar
Brazauskas, V. and Serfling, R. (2003) Favorable estimator for fitting Pareto models: A study using goodness-of-fit measures with actual data. Astin Bulletin, 33 (2), 365–381.CrossRefGoogle Scholar
Castellanos, D. (1988) The ubiquitous pi. Mathematics Magazine, 61, 67–98.CrossRefGoogle Scholar
Jacob, E. and Jayakumar, K. (2012) On half–Cauchy distribution and process. International Journal of Statistika and Mathematika, 3 (2), 77–81.Google Scholar
Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences. Hoboken, NJ: John Wiley & Sons.CrossRefGoogle Scholar
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2008) Loss Models: From Data to Decisions, 3rd ed.Wiley.CrossRefGoogle Scholar
Marshall, A.W. and Olkin, I. (1997) A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84 (3), 641–652.CrossRefGoogle Scholar
Mata, A. (2000) Princing excess of loss reinsurance with reinstatements. Astin Bulletin, 30 (2), 349–368.CrossRefGoogle Scholar
Nadarajah, S. and Bakar, S.A.A. (2014) New composite models for the Danish fire insurance data. Scandinavian Actuarial Journal, 2, 180–187.CrossRefGoogle Scholar
Rajan, S., Wang, S., Inkol, R. and Joyal, A. (2006) Efficient approximations for the arctangent function. IEEE Signal Processing Magazine, 23 (3), 108–111.CrossRefGoogle Scholar
Rizzo, M.L. (2009) New goodness-of-fit tests for Pareto distributions. Astin Bulletin, 39 (2), 691–715.CrossRefGoogle Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugel, J. (1999) Stochastic Processes for Insurance and Finance. John Wiley & Sons.CrossRefGoogle Scholar
Rytgaard, M. (1990) Estimation in the Pareto distribution. Astin Bulletin, 20 (2), 201–216.CrossRefGoogle Scholar
Sarabia, J.M. and Castillo, E. (2005) About a class of max–stable families with applications to income distributions. Metron, LXIII, 3, 505–527.Google Scholar
Scollnik, D.P.M. (2014) Regarding folded models and the paper by Brazauskas and Kleefeld (2011) Scandinavian Actuarial Journal, 2014 (3), 278–281.CrossRefGoogle Scholar
Scollnik, D.P.M. and Sun, C. (2012) Modeling with Weibull–Pareto models. North American Actuarial Journal, 16 (2), 260–272.CrossRefGoogle Scholar
Vuong, Q. (1989) Likelihood ratio tests for model selection and non-nested hypotheis. Econometrica, 50, 1–25.Google Scholar
Yang, H. (2004) Cramér–Lundberg asymptotics. In Encyclopedia of Actuarial Science. Wiley.Google Scholar