Vol. 8, No. 5, 2015

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The Weibull distribution and Benford's law

Victoria Cuff, Allison Lewis and Steven J. Miller

Vol. 8 (2015), No. 5, 859–874

Benford’s law states that many data sets have a bias towards lower leading digits (about 30% are 1s). It has numerous applications, from designing efficient computers to detecting tax, voter and image fraud. It’s important to know which common probability distributions are almost Benford. We show that the Weibull distribution, for many values of its parameters, is close to Benford’s law, quantifying the deviations. As the Weibull distribution arises in many problems, especially survival analysis, our results provide additional arguments for the prevalence of Benford behavior. The proof is by Poisson summation, a powerful technique to attack such problems.

Benford's law, Weibull distribution, digit bias, Poisson summation
Mathematical Subject Classification 2010
Primary: 60F05, 11K06
Secondary: 60E10, 42A16, 62E15, 62P99
Received: 31 July 2014
Revised: 19 October 2014
Accepted: 1 December 2014
Published: 28 September 2015

Communicated by John C. Wierman
Victoria Cuff
Department of Mathematics
Clemson University
Clemson, SC 29634
United States
Allison Lewis
Department of Mathematics
North Carolina State University
Raleigh, NC 27695
United States
Steven J. Miller
Department of Mathematics and Statistics
Williams College
Williamstown, MA 01267
United States