### Miller, SJ and Nigrini, MJ (2006)

#### Order Statistics and Shifted Almost Benford Behavior

Posted on Math Arxiv, January 13, 2006.

**ISSN/ISBN:** Not available at this time.
**DOI:** Not available at this time.

**Abstract:** Fix a base B and let ζ have the standard exponential distribution; the distribution of digits of ζ base B is known to be very close to Benford's Law. If there exists a C such that the distribution of digits of C times the elements of the system is the same as that of ζ, we say the system exhibits Shifted Almost Benford behavior base B (with a shift of log_{B}C mod 1). Let X_{1}, ..., X_{N} be independent identically distributed random variables. If the X_{i}'s are drawn from the uniform distribution [0,L], then as N tends to infinity the distribution of the digits of the differences between adjacent X_{i}'s converges to Shifted Almost Benford behavior (with a shift of log_{B} L/N).
Fix a δ∈(0,1) and choose N independent random variables from a nice probability density. The distribution of digits of any N^{δ}consecutive differences and all N-1 normalized differences of the X_{i}'s exhibit Shifted Almost Benford behavior. We derive conditions on the probability density which determine whether or not the distribution of the digits of all the un-normalized differences converges to Benford's Law, Shifted Almost Benford behavior, or oscillates between the two, and show that the Pareto distribution leads to oscillating behavior.

**Bibtex:**

```
@misc{,
title={Order statistics and shifted almost Benford behavior},
author={Miller, Steven J. and Nigrini, Mark},
year={2006},
url={http://arxiv.org/abs/math/0601344},
}
```

**Reference Type:** E-Print

**Subject Area(s):** Not specified