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Aki’s maximum likelihood method of Gutenberg–Richter b-value estimation is based on the premise that magnitudes above a given threshold are distributed exponentially and presupposes that sampling is adequate for all these magnitudes; clearly, sampling will not be adequate for magnitudes having average recurrence times longer than the observation time. Thus, for any given sample, there is a higher-magnitude completeness threshold. The importance of considering this higher-magnitude threshold is discussed, the effect of incomplete large-magnitude sampling is evaluated, and a method is given for correcting the observations.

On a global scale, large earthquakes are Poisson-distributed in time which implies that the inter-earthquake intervals are exponentially distributed. Thus, a simple-minded estimation of the most probable interval could conclude that the most probable occurrence time for a large earthquake would be now! This thesis is unsupported by observations. The apparent paradox is explained when characteristics of the interval cumulative distributions are explored for different seismicity rates.