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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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\textit{S. Dlugosz} and \textit{U. Müller-Funk}, Adv. Data Anal. Classif., ADAC 3, No. 3, 281--290 (2009; Zbl 1306.60002)

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### References:

[1] | Benford F (1938) The law of anomalous numbers. Proc Am Philos Soc 78: 551–572 · JFM 64.0555.03 |

[2] | Bolton RJ, Hand DJ (2002) Statistical fraud detection: a review (with discussion). Stat Sci 17(3): 235–255 · Zbl 1013.62115 |

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[4] | Kemeny JG, Snell JL (1976) Finite Markov Chains. Springer, NewYork |

[5] | Nigrini MJ, Mittermaier LJ (1997) The use of Benford’s law as an aid in analytical procedures: audit. A J Pract Theory 16(2): 52–67 |

[6] | Rosenthal JS (1995) Convergence rates for Markov chains. SIAM Rev 37(3): 387–405 · Zbl 0833.60069 |

[7] | Schatte P (1988) On mantisse distributions in computing and Benford’s law. J Inf Process Cyber EIK 24: 443–455 · Zbl 0662.65040 |

[8] | Scott PD, Fasli M (2001) Benford’s law: an empirical investigation and a novel explanation. Technical report. CSM Technical Report 349, Department of Computer Science, University of Essex, http://cswww.essex.ac.uk/technical-reports/2001/CSM-349.pdf |

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