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.

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

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.

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