  Rank-ordering statistics provides a perspective on the rare, largest elements
of a population, whereas the statistics of cumulative distributions are
dominated by the more numerous small events. The exponent of a power law
distribution can be determined with good accuracy by rank-ordering statistics
from the observation of only a few tens of the largest events. Using analytical
results and synthetic tests, we quantify the systematic and the random errors.
  We also study the case of a distribution defined by two branches, each having
a power law distribution, one defined for the largest events and the other for
smaller events, with application to the World-Wide (Harvard) and Southern
California earthquake catalogs. In the case of the Harvard moment catalog, we
make more precise earlier claims of the existence of a transition of the
earthquake magnitude distribution between small and large earthquakes; the
$b$-values are $b_2 = 2.3 \pm 0.3$ for large shallow earthquakes and $b_1 =
1.00 \pm 0.02$ for smaller shallow earthquakes. However, the cross-over
magnitude between the two distributions is ill-defined. The data available at
present do not provide a strong constraint on the cross-over which has a $50\%$
probability of being between magnitudes $7.1$ and $7.6$ for shallow
earthquakes; this interval may be too conservatively estimated. Thus, any
influence of a universal geometry of rupture on the distribution of earthquakes
world-wide is ill-defined at best. We caution that there is no direct evidence
to confirm the hypothesis that the large-moment branch is indeed a power law.
In fact, a gamma distribution fits the entire suite of earthquake moments from
the smallest to the largest satisfactorily. There is no evidence that the
earthquakes of the Southern California catalog have a distribution with two
