Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-22T20:15:34.349Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Use of Equispaced Discrete Distributions

Published online by Cambridge University Press:  29 August 2014

J.F. Walhin  and
J. Paris
J.F. Walhin*
Affiliation:
Institut de Statistique, Université Catholique de Louvain, Belgium Le Mans Assurances, Belgique
J. Paris*
Affiliation:
Institut de Statistique, Université Catholique de Louvain, Belgium
*
*Institut de Statistique, Voie du Roman Pays, 20 B-1348 Louvain-la-Neuve, Belgique
•Le Mans Assurances, Avenue Louise, 222 B-1050 Bruxelles, Belgique
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Kolmogorov distance is used to transform arithmetic severities into equispaced arithmetic severities in order to reduce the number of calculations when using algorithms like Panjer's formulae for compound distributions. An upper bound is given for the Kolmogorov distance between the true compound distribution and the transformed one. Advantages of the Kolmogorov distance and disadvantages of the total variation distance are discussed. When the bounds are too big, a Berry-Esseen result can be used. Then almost every case can be handled by the techniques described in this paper. Numerical examples show the interest of the methods.

Keywords

Recursive formulacompound distributioninfinite divisibilityequispaced discrete distributionKolmogorov distancetotal variation distanceupper and lower boundsBerry-Esseen bound
Type
Workshops
Information
ASTIN Bulletin: The Journal of the IAA , Volume 28 , Issue 2 , November 1998 , pp. 241 - 255
Copyright
Copyright © International Actuarial Association 1998

References

Barbour, A.D., Holst, L., and Janson, S. (1992). Poisson Approximation. Oxford University Press.CrossRefGoogle Scholar
De Pril, N. (1985). Recursions for Convolutions of Arithmetic Distributions. ASTIN Bulletin, 15: 135–139.CrossRefGoogle Scholar
De Pril, N. and Dhaene, J. (1992). Error Bounds for Compound Poisson Approximations of the Individual Risk Model. ASTIN Bulletin, 22: 137–148.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications Vol II (3ed). Wiley, New York.Google Scholar
Gerber, H.U. (1982). On the Numerical Evaluation of the Distribution of Aggregate Claims and its Stop-Loss Premiums, insurance: Mathematics and Economics, 1: 13–18.Google Scholar
Kestemont, R.M. and Paris, J. (1985). Sur l'Ajustement du Nombre de Sinistres. Bulletin of the Swiss Actuaries, 85: 157–163.Google Scholar
Michel, R. (1993). On Berry-Esseen Results for the Compound Poisson Distribution. Insurance: Mathematics and Economics, 13: 35–37.Google Scholar
Panjer, H.H. (1981). Recursive Evaluation of a Family of Compound Distributions. ASTIN Bulletin, 12: 22–26.CrossRefGoogle Scholar
Panjer, H.H. and Lutkk, B.W. (1983). Practical Aspects of Stop-Loss Calculations. Insurance: Mathematics and Economics, 2: 159–177.Google Scholar
Panjer, H.H. and Willmot, G.E. (1992). Insurance Risk Models. Society of Actuaries.Google Scholar
van Beek, P. (1972). An Application of Fourier Methods to the Problem of Sharpening the Berry-Esseen Inequality. Z. Wahrscheinlichkeitstheorie verw. Geb., 23: 187–196.CrossRefGoogle Scholar
Walhin, J.F. and Paris, J. (1998). A Large Family of Discrete Probability Laws, forthcoming.Google Scholar
Wang, S. and Panjer, H.H. (1994). Proportional Convergence and Tail-Cutting Techniques in Evaluating Aggregate Claim Distributions. Insurance: Mathematics and Economics, 14: 129–138.Google Scholar