  A wide range of natural and social phenomena result in observables whose
distributions can be well approximated by a power-law decay. The well-known
Hill estimator of the tail exponent provides results which are in many respects
superior to other estimators in case the asymptotics of the distribution is
indeed a pure power-law, however,systematic errors occur if the distribution is
altered by simply shifting it. We demonstrate some related problems which
typically emerge when dealing with empirical data and suggest a procedure
designed to extend the applicability of the Hill estimator.
