Correcting exponentiality test for binned earthquake magnitudes

Abstract

Above the magnitude of completeness - the minimum threshold for which a 100\% detection rate is assumed - earthquake magnitudes are typically modeled as a continuous exponential distribution. In practice, however, earthquake catalogs report magnitudes with finite resolution, resulting in a discrete (geometric) distribution. To determine the magnitude of completeness, 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. Here we show analytically that uniform dithering does not recover the underlying continuous exponential distribution from its discretized (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. Through numerical experiments, we demonstrate that this deviation yields a systematic overestimation of the magnitude of completeness, with biases exceeding one magnitude unit in large, high-resolution catalogs. We derive the exact noise distribution - a truncated exponential within each magnitude bin - that correctly restores the continuous exponential distribution over the whole magnitude range. Numerical tests show that this correction yields Lilliefors rejection probabilities that are consistent with the significance level across a wide range of bin widths and catalog sizes. Although illustrated for the Lilliefors test, the identified bias and the proposed correction are independent of the specific statistical test and apply generally to exponentiality testing of discretized magnitude data.