Monotonie des Hill-Schätzers in endlichen Stichproben

If the Pareto parameter Alpha of a Pareto-distributed random variable is calculated as Maximum-Likelihood estimator (Hill-estimator) from a random sample, the expected value of the estimator is a decreasing function of the sample size. In the present paper we show this property also to be valid, if a finite collection of positive real numbers is approximated by a Pareto-distribution, the Pareto parameter of which is calculated as Hill-estimator from samples of the collection. Using the Hill-estimator for Alpha, the expected value of the collection can be estimated as the expected value of the corresponding Pareto-distribution. Generally, the expected value of a Pareto-distributed random variable is a decreasing function of Alpha. Paradoxically however, if the mean value of the Pareto-distribution is used as an estimator for the expected value of the above collection, this estimator also proves to be a decreasing function of the sample size. This property is relevant for the quotation of reinsurance contracts.