International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 382948, 19 pages
Research Article

Order Statistics and Benford's Law

Steven J. Miller1 and Mark J. Nigrini2

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 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. Let X1,,XN be i.i.d.r.v. If the Xi's are Unif, then as N the distribution of the digits of the differences between adjacent order statistics converges to shifted exponential behavior. If instead Xi'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 Nδ consecutive differences and all N1 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.