Article,
Extreme-value based estimation of the conditional tail moment with application to reinsurance rating
Affiliations
- [1] Univ Southern Denmark, Dept Math & Comp Sci, Campusvej 55, DK-5230 Odense M, Denmark [NORA names: SDU University of Southern Denmark; University; Denmark; Europe, EU; Nordic; OECD];
- [2] Univ Strasbourg, Inst Rech Math Avancee, UMR 7501, 7 Rue Rene Descartes, F-67084 Strasbourg, France [NORA names: France; Europe, EU; OECD];
- [3] CNRS, 7 Rue Rene Descartes, F-67084 Strasbourg, France [NORA names: France; Europe, EU; OECD];
- [4] CNRS, 7 Rue Rene Descartes, F-67084 Strasbourg, France [NORA names: France; Europe, EU; OECD];
- [5] CNRS, 7 Rue Rene Descartes, F-67084 Strasbourg, France [NORA names: France; Europe, EU; OECD];
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Abstract
We study the estimation of the conditional tail moment, defined for a non-negative random variable X as theta(beta,p) = E (X-beta vertical bar X > U(1/p)), beta > 0, p is an element of (0, 1), provided E(X-beta) < infinity, where Udenotes the tail quantile function given by U(x) = inf{y : F(y) >= 1 - 1/x}, x > 1, associated to the distribution function F of X. The focus will be on situations where p is small, i.e., smaller than 1/n, where n is the number of observations on X that is available for estimation. This situation corresponds to extrapolation outside the data range, and requires extreme value arguments to construct an appropriate estimator. The asymptotic properties of the estimator, properly normalised, are established under suitable conditions. The developed methodology is applied to estimation of the expected payment and the variance of the payment under an excess-of-loss reinsurance contract. We examine the finite sample performance of the estimators by a simulation experiment and illustrate their practical use on the Secura Belgian Re automobile claim data. (C) 2022 Elsevier B.V. All rights reserved.