  Many natural processes exhibit power-law behavior. The power-law exponent is
linked to the underlying physical process and therefore its precise value is of
interest. With respect to the energy content of nanoflares, for example, a
power-law exponent steeper than 2 is believed to be a necessary condition to
solve the enigmatic coronal heating problem. Studying power-law distributions
over several orders of magnitudes requires sufficient data and appropriate
methodology. In this paper we demonstrate the shortcomings of some popular
methods in solar physics that are applied to data of typical sample sizes. We
use synthetic data to study the effect of the sample size on the performance of
different estimation methods and show that vast amounts of data are needed to
obtain a reliable result with graphical methods (where the power-law exponent
is estimated by a linear fit on a log-transformed histogram of the data). We
revisit published results on power laws for the angular width of solar coronal
mass ejections and the radiative losses of nanoflares. We demonstrate the
benefits of the maximum likelihood estimator and advocate its use.
