The Maximum Likelihood Estimator of b-Value for Mainshocks

Considering the magnitudes of events in a catalog as independent and exponentially distributed random variables, in agreement with the Gutenberg-Richter law, the statistical significance of the difference between the maximum likelihood estimators of the b-value for mainshocks (bms) and for all events (ball) is discussed. It is shown that, as a consequence of their definition, the mainshocks do not entirely satisfy the Gutenberg–Richter law and that bms has been frequently estimated as lower than ball because of an incorrect use of the maximum likelihood method.