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.

翻译：在完整性震级（即假定检测率达到100%的最小阈值）之上，地震震级通常被建模为连续指数分布。然而在实际操作中，地震目录报告的震级具有有限分辨率，从而形成离散（几何）分布。为确定完整性震级，通常采用Lilliefors检验。由于该检验假设数据连续，标准做法是在检验指数性前向分档震级添加均匀噪声。本文通过解析证明，均匀抖动并不能从其离散化（几何）形式恢复潜在的连续指数分布，而是返回一个分段恒定的剩余寿命分布——当目录规模或分档宽度增大时，该分布与指数模型的偏差将变得可检测。通过数值实验，我们证明这种偏差会导致对完整性震级的系统性高估，在大型高分辨率目录中偏差可超过一个震级单位。我们推导出精确的噪声分布——每个震级档内的截断指数分布——该分布能在整个震级范围内正确恢复连续指数分布。数值测试表明，此修正产生的Lilliefors拒绝概率在广泛的分档宽度和目录规模范围内与显著性水平保持一致。虽然以Lilliefors检验为例进行说明，但所识别的偏差及提出的修正与具体统计检验方法无关，普遍适用于离散化震级数据的指数性检验。