  We study the distributions of earthquake numbers in two global catalogs:
Global Centroid-Moment Tensor and Preliminary Determinations of Epicenters.
These distributions are required to develop the number test for forecasts of
future seismic activity rate. A common assumption is that the numbers are
described by the Poisson distribution. In contrast to the one-parameter Poisson
distribution, the negative-binomial distribution (NBD) has two parameters. The
second parameter characterizes the clustering or over-dispersion of a process.
We investigate the dependence of parameters for both distributions on the
catalog magnitude threshold and on temporal subdivision of catalog duration. We
find that for most cases of interest the Poisson distribution can be rejected
statistically at a high significance level in favor of the NBD. Therefore we
investigate whether these distributions fit the observed distributions of
seismicity. For this purpose we study upper statistical moments of earthquake
numbers (skewness and kurtosis) and compare them to the theoretical values for
both distributions. Empirical values for the skewness and the kurtosis increase
for the smaller magnitude threshold and increase with even greater intensity
for small temporal subdivision of catalogs. A calculation of the NBD skewness
and kurtosis levels shows rapid increase of these upper moments levels.
However, the observed catalog values of skewness and kurtosis are rising even
faster. This means that for small time intervals the earthquake number
distribution is even more heavy-tailed than the NBD predicts. Therefore for
small time intervals we propose using empirical number distributions
appropriately smoothed for testing forecasted earthquake numbers.
