Correcting the exponentiality test applied to binned earthquake magnitudes

The Lilliefors test is commonly applied to assess the exponentiality of earthquake magnitudes and, consequently, to estimate the minimum threshold above which seismic events are completely recorded (the completeness magnitude). In theory, the test assumes continuously distributed exponential data; however, real earthquake catalogs typically report magnitudes with finite resolution, resulting in a discrete (geometric) distribution. To address this mismatch, standard practice adds uniform noise to the data prior to testing for exponentiality. 

In this work, we analytically demonstrate that uniform dithering cannot recover the exponential distribution from its geometric counterpart. Instead, it produces a piecewise-constant residual lifetime distribution, whose deviation from the exponential model becomes increasingly detectable as the catalog size or bin width increases, as confirmed also by numerical experiments. We further prove that an exponential distribution truncated over the bin interval is the exact noise distribution required to correctly restore the continuous exponential distribution over the whole magnitude range. Numerical tests also show that this correction yields Lilliefors rejection rates consistent with the significance level for all bin widths and catalog sizes. 

Correcting the exponentiality test for binned magnitudes according to these results ensures a more reliable estimation of the completeness threshold, particularly in the case of high-resolution earthquake catalogs.