Parameter Uncertainty in Exponential Family Tail Estimation

The final version of this article has appeared as: Landsman Z, Tsanakas, A. (2012), 'Parameter uncertainty in exponential family tail estimation', ASTIN Bulletin, 42(1), p.123 - 152.

34 Pages Posted: 29 Jun 2011 Last revised: 3 Jan 2014

See all articles by Zinoviy Landsman

Zinoviy Landsman

University of Haifa, Department of Statistics

Andreas Tsanakas

Bayes Business School (formerly Cass), City, University of London

Date Written: October 26, 2011

Abstract

Actuaries are often faced with the task of estimating tails of loss distributions from just a few observations. Thus estimates of tail probabilities (reinsurance prices) and percentiles (solvency capital requirements) are typically subject to substantial parameter uncertainty. We study the bias and MSE of estimators of tail probabilities and percentiles, with focus on 1-parameter exponential families. Using asymptotic arguments it is shown that tail estimates are subject to significant positive bias. Moreover, the use of bootstrap predictive distributions, which has been proposed in the actuarial literature as a way of addressing parameter uncertainty, is seen to double the estimation bias. A bias corrected estimator is thus proposed. It is then shown that the MSE of the MLE, the parametric bootstrap and the bias corrected estimators only differ in terms of order $O(n^{-2})$, which provides decision-makers with some flexibility as to which estimator to use. The accuracy of asymptotic methods, even for small samples, is demonstrated exactly for the exponential and related distributions, while other 1-parameter distributions are considered in a simulation study. We argue that the presence of positive bias may be desirable in solvency capital calculations, though not necessarily in pricing problems.

Keywords: reinsurance pricing, VAR, parameter uncertainty, bias, bootstrap, exponential families

JEL Classification: C13, D8

Suggested Citation

Landsman, Zinoviy and Tsanakas, Andreas, Parameter Uncertainty in Exponential Family Tail Estimation (October 26, 2011). The final version of this article has appeared as: Landsman Z, Tsanakas, A. (2012), 'Parameter uncertainty in exponential family tail estimation', ASTIN Bulletin, 42(1), p.123 - 152., Available at SSRN: https://ssrn.com/abstract=1874043 or http://dx.doi.org/10.2139/ssrn.1874043

Zinoviy Landsman

University of Haifa, Department of Statistics ( email )

Haifa, 31905
Israel
+972-4-8249003 (Phone)

HOME PAGE: http://stat.haifa.ac.il/~landsman

Andreas Tsanakas (Contact Author)

Bayes Business School (formerly Cass), City, University of London ( email )

106 Bunhill Row
London, EC1Y 8TZ
United Kingdom

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