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.