Application to the Detection of Customs Fraud of the Goodness-of-fit Testing for the Newcomb-benford Law

KEYWORDS: " anti-fraud " , " international trade " , " multiple testing " , " outlier detection " , " significant digit distribution ". 1 Abstract The " first digit " law, also called the " leading digit " phenomenon, was discovered by Simon Newcomb, the well-known mathematician and astronomer, who firstly noticed that the leading digit 1 is more likely to occur than the leading digit 2, the leading digit 2 is more likely than the leading digit 3, and so on Newcomb(1881). This lead to a probabilistic distribution that, many years later, was independently rediscovered and publicized by Frank Benford Benford(1938). He also emphasized that potential applications cover many types of numerical data, ranging from death rates to stock prices, from baseball statistics to the area of lakes. Several empirical studies has shown how the Newcomb-Benford (NB) law is actually a sort of universal numeric rule. Variables coming from different fields were proven to closely fit the NB distribution for the first digit: astronomic distances, the numbers of Fibonacci sequence, the number of inhabitants of distinct areas, the volumes daily traded in stock market are only very few examples. In general, every set of numbers that represent the measurements of a phenomenon without having a built-in minimum or maximum and covering a range of values enough large, is likely to fit the NB distribution. In a probabilistic setting, a deep analysis of the NB law was first carried out by Hill(1995); see also the survey paper by Berger and Hill(2011a). Loosely