International Journal of Mathematics and Mathematical Sciences, Art. ID 382948.
ISSN/ISBN: 0161-1712 DOI: 10.1155/2008/382948
Abstract: Fix a base B > 1 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 some set is the same as that of ζ, we say that set exhibits shifted exponential 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 on [0;L], then as N→∞ the distribution of the digits of the differences between adjacent order statistics converges to shifted exponential behavior (with a shift of log_{B} L/N mod 1). By differentiating the cumulative distribution function of the logarithms modulo 1, applying Poisson Summation and then integrating the resulting expression, we derive rapidly converging explicit formulas measuring the deviations from Benford's Law. Fix a δ∈(0, 1) and choose N independent random variables from any compactly supported distribution with uniformly bounded first and second derivatives and a second order Taylor series expansion at each point. The distribution of digits of any N^{δ} consecutive differences and all N-1 normalized differences of the order statistics exhibit shifted exponential 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 exponential behavior, or oscillates between the two, and show that the Pareto distribution leads to oscillating behavior.
Bibtex:
@article {MR2461421,
AUTHOR = {Miller, Steven J. and Nigrini, Mark J.},
TITLE = {Order statistics and {B}enford's law},
JOURNAL = {Int. J. Math. Math. Sci.},
FJOURNAL = {International Journal of Mathematics and Mathematical
Sciences},
YEAR = {2008},
PAGES = {Art. ID 382948, 19},
ISSN = {0161-1712},
MRCLASS = {62G30 (62E10)},
MRNUMBER = {2461421 (2010c:62168)},
DOI = {10.1155/2008/382948},
URL = {http://dx.doi.org/10.1155/2008/382948},
}
Reference Type: Journal Article
Subject Area(s): Statistics