  We develop an efficient numerical scheme to solve accurately the set of
nonlinear integral equations derived previously in (Saichev and Sornette,
2007), which describes the distribution of inter-event times in the framework
of a general model of earthquake clustering with long memory. Detailed
comparisons between the linear and nonlinear versions of the theory and direct
synthetic catalogs show that the nonlinear theory provides an excellent fit to
the synthetic catalogs, while there are significant biases resulting from the
use of the linear approximation. We then address the suggestions proposed by
some authors to use the empirical distribution of inter-event times to obtain a
better determination of the so-called clustering parameter. Our theory and
tests against synthetic and empirical catalogs find a rather dramatic lack of
power for the distribution of inter-event times to distinguish between quite
different sets of parameters, casting doubt on the usefulness of this
statistics for the specific purpose of identifying the clustering parameter.
