Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-07-05T21:35:16.216Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimation in the Pareto Distribution

Published online by Cambridge University Press:  29 August 2014

Mette Rytgaard
Mette Rytgaard*
Affiliation:
Nordisk Reinsurance Company A/S, Copenhagen, Denmark
*
Nordisk Reinsurance Company, Grønningen 25, DK-1270 Copenhagen K, Denmark.
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.

In the present paper, different estimators of the Pareto parameter α will be proposed and compared to each others.

First traditional estimators of α as the maximum likelihood estimator and the moment estimator will be deduced and their statistical properties will be analyzed. It is shown that the maximum likelihood estimator is biased but it can easily be modified to an minimum-variance unbiased estimator of a. But still the coefficient of variance of this estimator is very large.

For similar portfolios containing same types of risks we will expect the estimated α-values to be at the same level. Therefore, credibility theory is used to obtain an alternative estimator of α which will be more stable and less sensitive to random fluctuations in the observed losses.

Finally, an estimator of the risk premium for an unlimited excess of loss cover will be proposed. It is shown that this estimator is a minimum-variance unbiased estimator of the risk premium. This estimator of the risk premium will be compared to the more traditional methods of calculating the risk premium.

Type
Workshop
Information
ASTIN Bulletin: The Journal of the IAA , Volume 20 , Issue 2 , November 1990 , pp. 201 - 216
Copyright
Copyright © International Actuarial Association 1990

References

Benktander, G. (1970) Schadenverteilung nach Grösse in der Nicht-lebenversicherung. Mitteilungen der Vereinigung Schweizerischer Versicherungsmathematiker, 70.Google Scholar
Benktander, G. (1978) Largest Claims Reinsurance (LCR). A Quick method to calculate LCR-Risk Rates from Excess of Loss Risk-rates. ASTIN Bulletin 10, 54–58.CrossRefGoogle Scholar
Benktander, G. (1988) Security Loadings in excess of loss. Transactions of the 23rd International Congress of Actuaries, Helsinki 4, 9–18.Google Scholar
Bühlmann, Hans (1970) Mathematical Methods in Risk Theory. Springer-Verlag, Berlin-Heidelberg.Google Scholar
Hogg, Robert V. and Klugman, Stuart A. (1984) Loss Distributions. John Wiley & Sons, New York.CrossRefGoogle Scholar
Mukhopadhyay, N. and Ekwo, M.E. (1987) Sequential Estimation Problems for the Scale Parameter of a Pareto Distribution. Scandinavian Actuarial Journal, 83–103.CrossRefGoogle Scholar
Rao, C. R. (1973) Linear Statitistical Inference and Its Applications. John Wiley & Sons, New York.CrossRefGoogle Scholar
Silvey, S.D. (1970) Statistical Inference. Penguin Books, Harmondsworth, Middlesex.Google Scholar