Abstract
Consider the sample X1, X2, ..., XN of N independent and identically distributed (iid) random variables with common cumulative distribution function (cdf)F, and let Fu be their conditional excess distribution function F. We define the ordered sample by . Pickands (1975), Balkema and de Haan (1974) posed that for a large class of underlying distribution functions F , and large u ,Fu is well approximated by the Generalized Pareto Distribution.The mixed method is a method for determining thresholds. This method consists in minimizing the variance of a convex combination of other thresholds.The objective of the mixed method is to determine by which probability distribution one can approach this conditional distribution. In this article, we propose a theorem which specifies the conditional distribution of excesses when the deterministic threshold tends to the end point.
Highlights
Pareto distribution is traditionally used by reinsurer’s excess of loss mainly because of its good mathematical properties, from the simplicity of the formulas resulting from its application
We propose a theorem which specifies the conditional distribution of excesses when the deterministic threshold tends to the end point
The new mixed method (MM) was proposed in [1, 2, 3, 4] to determine a threshold U = ∑pk=1 αk Uk + α3U3 with 1 ≤ p ≤ 2, at which a unit is declared atypical minimizing the variance of a convex combination of thresholds obtained by the mean excess function and generalized Pareto distribution
Summary
Pareto distribution is traditionally used by reinsurer’s excess of loss mainly because of its good mathematical properties, from the simplicity of the formulas resulting from its application. The new mixed method (MM) was proposed in [1, 2, 3, 4] to determine a threshold U = ∑pk=1 αk Uk + α3U3 with 1 ≤ p ≤ 2, at which a unit is declared atypical minimizing the variance of a convex combination of thresholds obtained by the mean excess function and generalized Pareto distribution (extreme quantile were estimated with a probability of 99.9% being an extreme value for the distribution of amounts of sinister with a confidence level of 95%). This method allows a compromi between the GPD method and FME method, between a minimum strategy GPD and maximum strategy FME (Mean Excess Function) It is more correlated with the GPD method and relatively smooth
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