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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.

MSC:

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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