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Many natural processes exhibit power-law behavior. The power-law exponent islinked to the underlying physical process and therefore its precise value is ofinterest. With respect to the energy content of nanoflares, for example, apower-law exponent steeper than 2 is believed to be a necessary condition tosolve the enigmatic coronal heating problem. Studying power-law distributionsover several orders of magnitudes requires sufficient data and appropriatemethodology. In this paper we demonstrate the shortcomings of some popularmethods in solar physics that are applied to data of typical sample sizes. Weuse synthetic data to study the effect of the sample size on the performance ofdifferent estimation methods and show that vast amounts of data are needed toobtain a reliable result with graphical methods (where the power-law exponentis estimated by a linear fit on a log-transformed histogram of the data). Werevisit published results on power laws for the angular width of solar coronalmass ejections and the radiative losses of nanoflares. We demonstrate thebenefits of the maximum likelihood estimator and advocate its use.

The Effect of Limited Sample Sizes on the Accuracy of the Estimated Scaling Parameter for Power-Law-Distributed Solar Data