International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 382948, 19 pages
Order Statistics and Benford's Law
1Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267, USA
2Accounting and Information Systems, School of Business, The College of New Jersey, Ewing, NJ 08628, USA
Received 2 June 2008; Revised 6 September 2008; Accepted 13 October 2008
Academic Editor: Jewgeni Dshalalow
Copyright © 2008 Steven J. Miller and Mark J. Nigrini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Fix a base and let have the standard exponential
distribution; the distribution of digits of base is
known to be very close to Benford's law. If there exists a such
that the distribution of digits of times the elements of some set is the same as that of , we say that set exhibits
shifted exponential behavior base . Let be i.i.d.r.v. If the 's are Unif, then as the distribution
of the digits of the differences between adjacent order statistics
converges to shifted exponential behavior. If instead 's come from a compactly supported distribution with uniformly bounded first and second derivatives and a second-order Taylor series expansion at each point, then the distribution of digits of any consecutive differences and all 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
unnormalized differences converges to Benford's law, shifted exponential behavior, or oscillates between the two, and show that the Pareto distribution leads to oscillating behavior.