Loi de Benford générale

摘要

A variable X satisfies the “Benford law” if log(X) has a uniform distribution modulo 1. This law approximately applies to many experimental or observational data sets.Many theories have been put forward as explanations for this phenomenon, mostly based on the characteristics of the log function. An elementary new explanation has recently been published, based on the fact that any X whose distribution is “smooth” and “scattered” enough is Benford. The scattering and smoothness of usual data ensures that log(X) is itself smooth and scattered, which in turn implies the Benford characteristic of X.If this explanation is the good one, the Benford law should not depend on the log function itself. In this paper, we define and test a “General Benford Law” for a function u. X satisfies this law if u(X) is uniform modulo 1. Statistical data, mathematical series and continuous variables are tested for functions log(log), square, square root. The results suggest that the Benford law for function log is not pathological, and that other functions also apply to natural data. We discuss possible interests and properties of this general Benford law.