Asymptotic normality of least-squares estimators of tail indices

Based on least-squares considerations, Schultze and Steinebach proposed three new estimators for the tail index of a regularly varying distribution function and proved their consistency. We show that, unlike the 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 desirable order of the norming sequence. We analyse the conditions under which asymptotic confidence intervals become possible. In a submodel, we compare the asymptotic mean square errors of optimal versions of these and earlier estimators. The choice of the number of extreme order statistics to be used is also discussed through the investigation of the asymptotic mean square error for a comprehensive set of examples of a general kind.