  The size or energy of diverse structures or phenomena in geoscience appears
to follow power-law distributions. A rigorous statistical analysis of such
observations is tricky, though. Observables can span several orders of
magnitude, but the range for which the power law may be valid is typically
truncated, usually because the smallest events are too tiny to be detected and
the largest ones are limited by the system size.
  We revisit several examples of proposed power-law distributions dealing with
potentially damaging natural phenomena. Adequate fits of the distributions of
sizes are especially important in these cases, given that they may be used to
assess long-term hazard. After reviewing the theoretical background for
power-law distributions, we improve an objective statistical fitting method and
apply it to diverse data sets. The method is described in full detail and it is
easy to implement.
  Our analysis elucidates the range of validity of the power-law fit and the
corresponding exponent, and whether a power-law tail is improved by a truncated
log-normal. We confirm that impact fireballs and Californian earthquakes show
untruncated power-law behavior, whereas global earthquakes follow a double
power law. Rain precipitation over space and time and tropical cyclones show a
truncated power-law regime. Karst sinkholes and wildfires, in contrast, are
better described by truncated log-normals, although wildfires also may show
power-law regimes. Our conclusions only apply to the analyzed data sets, but
show the potential of applying this robust statistical technique in the future.
