Miller, S. J. (ed.): Benford’s law. Theory and applications

Abstract

Ever since its rediscovery in the late 1930s byFrankA.Benford, the strangedistribution of first digits observed in many data sets has become an increasingly attractive object of investigation in mathematics, statistics, economics, accounting and many other fields. And it is from this universe of applications and inspirations that the 19 chapters of this nicely edited survey are drawn. I particularly liked passages like the one by Frank A. Benford’s grandson (also named Frank A. Benford) on the history of the subject, or personal accounts where authors explain how their interest in Benford’s law arose in the first place. Then there are many chapters on the law’s most popular application, i.e. how Benford’s law can be used to detect fraud in data sets of various sorts, with an in depth treatment given to election outcomes, economic macrodata, or financial statements of all sorts. Although most of these applications have long been known and have also been widely discussed in the media, in particular those concerning fraud in tax returns, it is nice to have them all collected together in one place. More mathematically inclined readers will also appreciate the collection of mathematical explanations for Benford’s law in various probability distributions and stochastic processes. And for statisticians, there is also some discussion on how the significance of deviations from Benford’s law can be best determined. This diversity of authors and approaches almost by necessity engenders considerable variation in mathematical rigour and level of abstraction. It also leads to considerable overlap at various places. For instance, after reading for the umpteenth time that Benford detected his law by studying tables of logarithms, and that the relative frequency of the first digit 1 is equal to log 2, some weariness arises. Because of this diversity, I also very much doubt whether this book is useful as a textbook for advanced statistics courses,