Benford's Law: Hammering a Square Peg into a Round Hole?

Many authors have discussed the reasons why Benford's distribution for the most significant digits is seemingly so widespread. However the discussion is not settled because there is no theorem explaining its prevalence, in particular for naturally occurring scale-invariant data. Here we review Benford's distribution for continuous random variables under scale invariance. The implausibility of strict scale invariance leads us to a generalisation of Benford's distribution based on Pareto variables. This new model is more realistic, because real datasets are more prone to complying with a relaxed, rather than strict, definition of scale invariance. We also argue against forensic detection tests based on the distribution of the most significant digit. To show the arbitrariness of these tests, we give discrete distributions of the first coefficient of a continued fraction which hold in the exact same conditions as Benford's distribution and its generalisation.