  It has been repeatedly stated that maximum likelihood (ML) estimates of
exponents of power-law distributions can only be reliably obtained for
exponents smaller than minus one. The main argument that power laws are
otherwise not normalizable, depends on the underlying sample space the data is
drawn from, and is true only for sample spaces that are unbounded from above.
Here we show that power-laws obtained from bounded sample spaces (as is the
case for practically all data related problems) are always free of such
limitations and maximum likelihood estimates can be obtained for arbitrary
powers without restrictions. Here we first derive the appropriate ML estimator
for arbitrary exponents of power-law distributions on bounded discrete sample
spaces. We then show that an almost identical estimator also works perfectly
for continuous data. We implemented this ML estimator and discuss its
performance with previous attempts. We present a general recipe of how to use
these estimators and present the associated computer codes.
