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Parameter Uncertainty in Exponential Family Tail Estimation

Landsman, Z. & Tsanakas, A. (2012). Parameter Uncertainty in Exponential Family Tail Estimation. ASTIN Bulletin, 42(1), pp. 123-152. doi: 10.2143/AST.42.1.2160738


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.

Publication Type: Article
Additional Information: Copyright Cambridge University Press 2012. This version may have been revised following peer review but may be subject to further editorial input by Cambridge University Press.
Publisher Keywords: Reinsurance pricing; VaR; parameter uncertainty; bias; bootstrap; exponential families
Subjects: H Social Sciences > HF Commerce
Departments: Bayes Business School > Actuarial Science & Insurance
Related URLs:
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