# Implications of a recent result on Benford's law

I want to the discuss the implications of a theorem by J. Morrow (2010) regarding Benford's law.

There are many papers written about Benford's law with a comprenhensive discussion of the advantages and perils of using Benford's law to detect anaomalous data. A nice compilation up to 2006 can be found here.

One of the reasons Benford's law is not always very useful in "real" life is that we don't know when a database should follow Benford's distribution (even afer T. Hill's work on the subject). A lot of work has been done exemplifying this databses and giving empirical suggestions on when to use it. There has also been some experimental analysis (e.g., here) showing human attempts to fake random data showing they do not follow Benford's law in many cases. What seems to me very dissapointing though is that not even data sampled from a simple standard normal distributions would follow Benford's law very closely.

The theorem in J. Morrow's paper quoted above stablishes the following result:

Theorem. Let $X$ be a random variable with continuous pdf then for every $\epsilon > 0$ there is an $\alpha^*$ such that for all $\alpha \geq \alpha^*$: $$(X/\sigma)^{\alpha} \quad \epsilon\text{-satisfies Benford's law for all } \sigma \neq 0$$

Definition. A random variable $X$ is said to $\epsilon$-satisfy Benford's law if $$\left\lvert \mathbb{P}(X \text{ has first-significant-digit } d) - \log \left( 1+\frac{1}{d} \right) \right\rvert < \epsilon$$

It turns out that know many random variables should follow Benford's law after a suitable power transformation. I have run this on many real life databases and a lot of them seem now to converge to Benford's law predicted digital distribution. I have been thinking a lot about the applicability of this result. I have the following questions.

1. What could be the consequences of this results to study anomalous data?

2. If I take some data, transform it with a very high power and then verify its conformity to Benford's distribution, can I really conclude something of the original data?

Any ideas, suggestions or particular bibliography is highly appreciated.

• Problem is, faked data will also $\epsilon$-satisfy Benford's laws unless it was generated in a very specific way which fraudsters are unlikely to use because it transforms both "real" data and fake data to similar distributions on the first digit. – Dan Piponi Sep 10 '13 at 20:42
• I don't really have anything meaningful to say about this problem except that I find it interesting and I hope you get a good answer. The only questions that spring to my mind after having read your question (and recently having thought about Benford's Law while teaching a stats class) are: 1) can you find some asymptotics describing the relationship between $\epsilon$ and $\alpha^*$? and 2) are there any formal properties which anomalous data satisfies? Related to 2 is the question of "what is truly random" and perhaps papers by Jim Propp might help. – David White Sep 10 '13 at 21:49
• The answer to Question 2 would seem to be that all you can conclude is that maybe the original data had a continuous pdf. – Gerry Myerson Sep 10 '13 at 23:02
• @GerryMyerson or maybe be that it is not random? That would be more helpful. – Mauricio G Tec Sep 11 '13 at 2:39
• Although is not clear what does "not random" even means. – Mauricio G Tec Sep 11 '13 at 2:41