Astin Bulletin · 1990 · 125 citations · 3 references
Natural SciencesMathematicsProbability TheoryStatisticsStochastic ProcessesDensity EstimationEstimation TheoryInferential StatisticsLikelihood EstimationProbability DistributionsStatistical InferenceParametric EstimationPareto Optimal SolutionPosterior Distribution Estimation
Similar portfolios with the same risk types are expected to yield comparable estimated α-values. The paper proposes and compares several estimators of the Pareto parameter α and introduces an estimator for the risk premium of unlimited excess‑of‑loss coverage. The authors derive maximum‑likelihood and moment estimators of α, analyze their properties, use credibility theory to obtain a more stable α estimator, and compare a new risk‑premium estimator to traditional methods. The maximum‑likelihood estimator is biased but can be adjusted to a minimum‑variance unbiased estimator of α, though its coefficient of variation remains large, and the proposed risk‑premium estimator is also minimum‑variance unbiased.
Abstract 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.
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Mathematical Methods in Risk Theory.
R. E. Beard, H. Buhlmann · Journal of the Royal Statistical Society Series A (General) · 1971 · 508 citations
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