Skip to main content Accessibility help
×
Home
Hostname: page-component-5959bf8d4d-4p99k Total loading time: 0.2 Render date: 2022-12-08T10:32:24.834Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

MODELLING INSURANCE DATA WITH THE PARETO ARCTAN DISTRIBUTION

Published online by Cambridge University Press:  19 June 2015

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

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.

Type
Research Article
Copyright
Copyright © Astin Bulletin 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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, 151158.CrossRefGoogle Scholar
Bozdogan, H. (1987) Model selection and Akaike's Information Criterion (AIC): The general theory and its analytical extensions. Psychometrika, 52 (3), 345370.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), 5974.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), 753757.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), 365381.CrossRefGoogle Scholar
Castellanos, D. (1988) The ubiquitous pi. Mathematics Magazine, 61, 6798.CrossRefGoogle Scholar
Jacob, E. and Jayakumar, K. (2012) On half–Cauchy distribution and process. International Journal of Statistika and Mathematika, 3 (2), 7781.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), 641652.CrossRefGoogle Scholar
Mata, A. (2000) Princing excess of loss reinsurance with reinstatements. Astin Bulletin, 30 (2), 349368.CrossRefGoogle Scholar
Nadarajah, S. and Bakar, S.A.A. (2014) New composite models for the Danish fire insurance data. Scandinavian Actuarial Journal, 2, 180187.CrossRefGoogle Scholar
Rajan, S., Wang, S., Inkol, R. and Joyal, A. (2006) Efficient approximations for the arctangent function. IEEE Signal Processing Magazine, 23 (3), 108111.CrossRefGoogle Scholar
Rizzo, M.L. (2009) New goodness-of-fit tests for Pareto distributions. Astin Bulletin, 39 (2), 691715.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), 201216.CrossRefGoogle Scholar
Sarabia, J.M. and Castillo, E. (2005) About a class of max–stable families with applications to income distributions. Metron, LXIII, 3, 505527.Google Scholar
Scollnik, D.P.M. (2014) Regarding folded models and the paper by Brazauskas and Kleefeld (2011) Scandinavian Actuarial Journal, 2014 (3), 278281.CrossRefGoogle Scholar
Scollnik, D.P.M. and Sun, C. (2012) Modeling with Weibull–Pareto models. North American Actuarial Journal, 16 (2), 260272.CrossRefGoogle Scholar
Vuong, Q. (1989) Likelihood ratio tests for model selection and non-nested hypotheis. Econometrica, 50, 125.Google Scholar
Yang, H. (2004) Cramér–Lundberg asymptotics. In Encyclopedia of Actuarial Science. Wiley.Google Scholar
12
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

MODELLING INSURANCE DATA WITH THE PARETO ARCTAN DISTRIBUTION
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

MODELLING INSURANCE DATA WITH THE PARETO ARCTAN DISTRIBUTION
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

MODELLING INSURANCE DATA WITH THE PARETO ARCTAN DISTRIBUTION
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *