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New Goodness-of-Fit Tests for Pareto Distributions*

Published online by Cambridge University Press:  09 August 2013

Maria L. Rizzo
Maria L. Rizzo*
Affiliation:
Dept. of Mathematics & Statistics, Bowling Green State University, Bowling Green, OH 43403, E-Mail: [email protected], Phone: 419-372-7474, Fax: 419-372-6092
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Abstract

A new approach to goodness-of-fit for Pareto distributions is introduced. Based on Euclidean distances between sample elements, the family of statistics and tests is indexed by an exponent in (0,2) on Euclidean distance. The corresponding tests are statistically consistent and have excellent performance when applied to heavy-tailed distributions. The exponent can be tailored to the particular Pareto distribution. The goodness-of-fit statistic measures all types of differences between distributions, hence it is also applicable as a minimum distance estimator. Implementation of the test statistics is developed and applied to estimation of the tail index in three well known examples of claims data, and compared with the classical EDF statistics.

Keywords

Paretogoodness-of-fitheavy-tailGiniclaims
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 39 , Issue 2 , November 2009 , pp. 691 - 715
Copyright
Copyright © International Actuarial Association 2009

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Footnotes

*

This research was supported by the Casualty Actuarial Society through The Actuarial Foundation 2008 Individual Grants Program.

References

Arnold, B.C. (1983) Pareto Distributions. International Co-operative Publishing House, Fairland, MD.Google Scholar
Baxter, M.A. (1980) Minimum variance unbiased estimation of the parameters of the Pareto distribution. Metrika, 27: 133–138.CrossRefGoogle Scholar
Beirlant, J., Teugels, J.L. and Vynckier, P. (1996) Practical Analysis of Extreme Values. Leuven University Press, Leuven, Belgium.Google Scholar
Brazauskas, V. and Serfling, R. (2000a) Robust and efficient estimation of the tail index of a single-parameter Pareto distribution. North American Actuarial Journal, 4(4): 12–27.Google Scholar
Brazauskas, V. and Serfling, R. (2000b) Robust estimation of tail parameters for two-parameter Pareto and exponential models via generalized quantile statistics. Extremes, 3(3): 231–249.Google Scholar
Brazauskas, V. and Serfling, R. (2001) Small sample performance of robust estimators of tail parameters for Pareto and exponential models. Journal of Statistical Computation and Simulation, 70(1): 1–19.Google Scholar
Brazauskas, V. and Serfling, R. (2003) Favorable estimators for fitting Pareto models: A study using goodness-of-fit measures with actual data. ASTIN Bulletin, 33(2): 365–381. DOI: 10.2143/AST.33.2.503698.CrossRefGoogle Scholar
Hoeffding, W. (1948) A class of statistics with asymptotically normal distribution. Annals of Mathematical Statistics, 19: 293–325.Google Scholar
Hogg, R.V. and Klugman, S.A. (1984) Loss Distributions. Wiley, New York.CrossRefGoogle Scholar
Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences. Wiley.CrossRefGoogle Scholar
Likeš, J. (1969) Minimum variance unbiased estimation of the parameters of power-function and Pareto's distribution. Statistische Hefte, 10: 104–110.Google Scholar
Patrik, G. (1980) Estimating casualty insurance loss amount distributions. Proceedings of the Casualty Actuarial Society, LXXVII: 57–109.Google Scholar
Philbrick, S.W. and Jurschak, J. (1981) Discussion of “Estimating casualty insurance loss amount distributions”. Proceedings of the Casualty Actuarial Society, LXVIII: 101–106.Google Scholar
Porter, J.E. III, Coleman, J.W. and Moore, A.H. (1992) Modified KS, AD, and C-vM tests for the Pareto distribution with unknown location & scale parameters. IEEE Transactions on Reliability, 41(1): 112–117.Google Scholar
Prudnikov, A.P., Brychkov, Y.A. and Marichev, O.I. (1990) Integrals and series (Integraly i riady, translated from the Russian by N.M. Queen). Gordon and Breach Science Publishers, New York.Google Scholar
Rytgaard, M. (1990) Estimation in the Pareto distribution. ASTIN Bulletin, 20(2): 201–216.Google Scholar
Stephens, M.A. (1986) Tests based on EDF statistics. In D'Agostino, R.B. and Stephens, M.A., editors, Goodness-of-Fit Techniques, pages 97–193. Marcel Dekker, New York.Google Scholar
Székely, G.J. and Rizzo, M.L. (2005a) A new test for multivariate normality. Journal of Multivariate Analysis, 93(1): 58–80.Google Scholar
Székely, G.J. and Rizzo, M.L. (2005b) Hierarchical clustering via joint between-within distances: extending Ward's minimum variance method. Journal of Classification, 22(2): 151–183.CrossRefGoogle Scholar
von Mises, R. (1947) On the asymptotic distributions of differentiable statistical functionals. Annals of Mathematical Statistics, 2: 209–348.Google Scholar