Goedele Dierckx

Tail Index Estimation and an Exponential Regression Model

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.

On Exponential Representations of Log-Spacings of Extreme Order Statistics

In Beirlant et al. (1999) and Feuerverger and Hall (1999) an exponential regression model (ERM) was introduced on the basis of scaled log-spacings between subsequent extreme order statistics from a Pareto-type distribution. This lead to the construction of new bias-corrected estimators for the tail index. In this note, under quite general conditions, asymptotic justification for this regression model is given as well as for resulting tail index estimators. Also, we discuss diagnostic methods for adaptive selection of the threshold when using the Hill (1975) estimator which follow from the ERM approach. We show how the diagnostic presented in Guillou and Hall (2001) is linked to the ERM, while a new proposal is suggested. We also provide some small sample comparisons with other existing methods.

An asymptotically unbiased minimum density power divergence estimator for the Pareto-tail index

We introduce a robust and asymptotically unbiased estimator for the tail index of Pareto-type distributions. The estimator is obtained by fitting the extended Pareto distribution to the relative excesses over a high threshold with the minimum density power divergence criterion. Consistency and asymptotic normality of the estimator is established under a second order condition on the distribution underlying the data, and for intermediate sequences of upper order statistics. The finite sample properties of the proposed estimator and some alternatives from the extreme value literature are evaluated by a small simulation experiment involving both uncontaminated and contaminated samples.

Trend Analysis of Extreme Values

In Dierckx and Teugels (Environmetrics 2:1–26) we concentrated on testing whether an instantaneous change occurs in the value of the extreme value index. This short article illustrates with an explicit example that in some cases the extreme value index seems to change gradually rather than instantaneously. To this end a moving Hill estimator is introduced. Further a change point analysis and a trend analysis are performed. With this last analysis it is investigated whether a linear trend appears in the extreme value index.