ASYMPTOTIC NORMALITY OF LEAST-SQUARES ESTIMATORS OF TAIL INDEXES

Based on least-squares considerations, Schultze and Steinebach propose d three new estimators for the tail index of a regularly varying distr ibution function and proved their consistency. We show that, unlike th e Hill estimator, all three least-squares estimators can be centred to have normal asymptotic distributions universally over the whole model , and for two of these estimators this in fact happens at the desirabl e order of the norming sequence. We analyse the conditions under which asymptotic confidence intervals become possible. In a submodel, we co mpare the asymptotic mean square errors of optimal versions of these a nd earlier estimators. The choice of the number of extreme order stati stics to be used is also discussed through the investigation of the as ymptotic mean square error for a comprehensive set of examples of a ge neral kind.