, Volume 2, Issue 2, pp 177-200

Tail Index Estimation and an Exponential Regression Model

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One of the most important problems involved in the estimation of Pareto indices is the reduction of bias in case the slowly varying part of the Pareto type model disappears at a very slow rate. In other cases, when the bias problem is not so severe, the application of well-known estimators such as the Hill (1975) and the moment estimator (Dekkers et al. (1989)) still asks for an adaptive selection of the sample fraction to be used in such estimation procedures. We show that in both circumstances, solutions can be constructed for the given problems using maximum likelihood estimators based on a regression model for upper order statistics. Via this technique one can also infer about the bias-variance trade-off for a given data set. The behavior of the new maximum likelihood estimator is illustrated through simulation experiments, among others for ARCH processes.