The value of the last digit: statistical fraud detection with digit analysis. (English) Zbl 1306.60002

Summary: Digit distributions are a popular tool for the detection of tax payers’ noncompliance and other fraud. In the early stage of digital analysis, M. J. Nigrini and L. I. Mittermaier [“The use of Benford’s law as an aid in analytical procedures”, A: J. Pract. Theory 16, No. 2, 52–67 (1997)] made use of F. Benford’s Law [Proc. Am. Philos. Soc. 78, 551–572 (1938; Zbl 0018.26502)] as a natural reference distribution. A justification of that hypothesis is only known for multiplicative sequences [P. Schatte, J. Inf. Process. Cybern. 24, No. 9, 443–455 (1988; Zbl 0662.65040)]. In applications, most of the number generating processes are of an additive nature and no single choice of ‘an universal first-digit law’ seems to be plausible [P. Scott and M. Fasli, “Benford’s law: an empirical investigation and a novel explanation”, Preprint (2001), http://cswww.essex.ac.uk/technical-reports/2001/CSM-349.pdf]. In that situation, some practioners (e.g. financial authorities) take recourse to a last digit analysis based on the hypothesis of a Laplace distribution. We prove that last digits are approximately uniform for distributions with an absolutely continuous distribution function. From a practical perspective, that result, of course, is only moderately interesting. For that reason, we derive a result for ‘certain’ sums of lattice-variables as well. That justification is provided in terms of stationary distributions.


62E10 Characterization and structure theory of statistical distributions
62P20 Applications of statistics to economics
91B99 Mathematical economics
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
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