The value of the last digit: statistical fraud detection with digit analysis

Regular Article

Abstract

Digit distributions are a popular tool for the detection of tax payers’ noncompliance and other fraud. In the early stage of digital analysis, Nigrini and Mittermaier (A J Pract Theory 16(2):52–67, 1997) made use of Benford’s Law (Benford in Am Philos Soc 78:551–572, 1938) as a natural reference distribution. A justification of that hypothesis is only known for multiplicative sequences (Schatte in J Inf Process Cyber EIK 24:443–455, 1988). 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 (Scott and Fasli in Benford’s law: an empirical investigation and a novel explanation. CSM Technical Report 349, Department of Computer Science, University of Essex, http://cswww.essex.ac.uk/technical-reports/2001/CSM-349.pdf, 2001). In that situation, some practioneers (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.

Keywords

Fraud detection Last digits Digit analysis Benford’s law 

Mathematics Subject Classification (2000)

60B10 62P20 91B99 

References

  1. Benford F (1938) The law of anomalous numbers. Proc Am Philos Soc 78: 551–572Google Scholar
  2. Bolton RJ, Hand DJ (2002) Statistical fraud detection: a review (with discussion). Stat Sci 17(3): 235–255MATHCrossRefMathSciNetGoogle Scholar
  3. Gray RM (2006) Toeplitz and circulant matrices: a review. Found Trends Commun Inf Theory 2(3): 155–239CrossRefGoogle Scholar
  4. Kemeny JG, Snell JL (1976) Finite Markov Chains. Springer, NewYorkMATHGoogle Scholar
  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): 52–67Google Scholar
  6. Rosenthal JS (1995) Convergence rates for Markov chains. SIAM Rev 37(3): 387–405MATHCrossRefMathSciNetGoogle Scholar
  7. Schatte P (1988) On mantisse distributions in computing and Benford’s law. J Inf Process Cyber EIK 24: 443–455MATHMathSciNetGoogle Scholar
  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

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.ZEWMannheimGermany
  2. 2.ERCISMünsterGermany

Personalised recommendations