Correcting exponentiality test for binned earthquake magnitudes

In theory, earthquake magnitudes follow an exponential distribution. In practice, however, earthquake catalogs report magnitudes with finite resolution, resulting in a discrete (geometric) distribution. To determine the lowest magnitude above which seismic events are completely recorded, the Lilliefors test is commonly applied. Because this test assumes continuous data, it is standard practice to add uniform noise to binned magnitudes prior to testing exponentiality. This work shows analytically that uniform dithering cannot recover the exponential distribution from its geometric form. It instead returns a piecewise-constant residual lifetime distribution, whose deviation from the exponential model becomes detectable as catalog size or bin width increases. Numerical experiments confirm that this deviation yields an overestimation of the magnitude of completeness in large catalogs. We therefore derive the exact noise distribution - a truncated exponential on the bin interval - that correctly restores the continuous exponential distribution over the whole magnitude range. Numerical tests show that this correction yields Lilliefors rejection rates consistent with the significance level for all bin widths and catalog sizes. The proposed solution eliminates a methodological bias in completeness estimation, which especially impacts high-resolution catalogs.