First Significant Digit Patterns From Mixtures of Uniform Distributions

Abstract

Traditional tests searching for human influence in data assume that, barring such influence, first significant digits (FSD) are uniformly distributed. More recent tests rely on Benford's law, postulating that lower digits are more likely than higher ones. I show that both patterns belong to a family arising from mixtures of uniforms, and characterize the FSD patterns for a one-parameter subset of the family. I also show that all family members exhibit decreasing FSD probabilities. The empirical analysis suggests that although the uniform FSD pattern and Benford's law are reasonable models for some data, alternative family members better fit other data.