4,080
Views
8
CrossRef citations to date
0
Altmetric
Theory and Methods

On Characterizations and Tests of Benford’s Law

, , &
Pages 1887-1903 | Received 13 Feb 2020, Accepted 02 Feb 2021, Published online: 22 Apr 2021

References

  • Allaart, P. C. (1997), “An Invariant-Sum Characterization of Benford’s Law,” Journal of Applied Probability, 34, 288–291. DOI: 10.2307/3215195.
  • Barabesi, L., Cerasa, A., Cerioli, A., and Perrotta, D. (2016a), “A New Family of Tempered Distributions,” Electronic Journal of Statistics, 10, 3871–3893. DOI: 10.1214/16-EJS1214.
  • Barabesi, L., Cerasa, A., Cerioli, A., and Perrotta, D. (2018), “Goodness-of-Fit Testing for the Newcomb-Benford Law With Application to the Detection of Customs Fraud,” Journal of Business and Economic Statistics, 36, 346–358.
  • Barabesi, L., Cerasa, A., Perrotta, D., and Cerioli, A. (2016b), “Modeling International Trade Data With the Tweedie Distribution for Anti-Fraud and Policy Support,” European Journal of Operational Research, 248, 1031–1043. DOI: 10.1016/j.ejor.2015.08.042.
  • Barabesi, L., and Pratelli, L. (2020), “On the Generalized Benford Law,” Statistics and Probability Letters, 160, Article 108702. DOI: 10.1016/j.spl.2020.108702.
  • Berger, A., and Hill, T. P. (2011), “Benford’s Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem,” Mathematical Intelligencer, 33, 85–91. DOI: 10.1007/s00283-010-9182-3.
  • Berger, A., and Hill, T. P. (2015), An Introduction to Benford’s Law, Princeton, NJ: Princeton University Press.
  • Berger, A., and Hill, T. P. (2020), “The Mathematics of Benford’s Law: A Primer,” Statistical Methods and Applications, DOI: 10.1007/s10260-020-00532-8.
  • Berger, A., and Twelves, I. (2018), “On the Significands of Uniform Random Variables,” Journal of Applied Probability, 55, 353–367. DOI: 10.1017/jpr.2018.23.
  • Bolton, R. J., and Hand, D. J. (2002), “Statistical Fraud Detection: A Review,” Statistical Science, 17, 235–255. DOI: 10.1214/ss/1042727940.
  • Cerasa, A., Torti, F., and Perrotta, D. (2016), “Heteroscedasticity, Multiple Populations and Outliers in Trade Data,” in Topics on Methodological and Applied Statistical Inference, eds. T. Di Battista, E. Moreno, and W. Racugno, Cham: Springer, pp. 43–50.
  • Cerioli, A. (2010), “Multivariate Outlier Detection With High-Breakdown Estimators,” Journal of the American Statistical Association, 105, 147–156. DOI: 10.1198/jasa.2009.tm09147.
  • Cerioli, A., Barabesi, L., Cerasa, A., Menegatti, M., and Perrotta, D. (2019), “Newcomb-Benford Law and the Detection of Frauds in International Trade,” Proceedings of the National Academy of Sciences of the United States of America, 116, 106–115. DOI: 10.1073/pnas.1806617115.
  • Cerioli, A., Farcomeni, A., and Riani, M. (2019), “Wild Adaptive Trimming for Robust Estimation and Cluster Analysis,” Scandinavian Journal of Statistics, 46, 235–256. DOI: 10.1111/sjos.12349.
  • Cerioli, A., and Perrotta, D. (2014), “Robust Clustering Around Regression Lines With High Density Regions,” Advances in Data Analysis and Classification, 8, 5–26. DOI: 10.1007/s11634-013-0151-5.
  • Diaconis, P. (1977), “The Distribution of Leading Digits and Uniform Distribution mod 1,” The Annals of Probability, 5, 72–81. DOI: 10.1214/aop/1176995891.
  • Diaconis, P., and Freedman, D. (1979), “On Rounding Percentages,” Journal of the American Statistical Association, 74, 359–364.
  • Dümbgen, L., and Leuenberger, C. (2008), “Explicit Bounds for the Approximation Error in Benford’s Law,” Electronic Communications in Probability, 13, 99–112.
  • Dümbgen, L., and Leuenberger, C. (2015), “Explicit Error Bounds via Total Variation,” in Benford’s Law: Theory and Applications, ed. S. J. Miller, Princeton, NJ: Princeton University Press, pp. 119–134.
  • Engel, H. A., and Leuenberger, C. (2003), “Benford’s Law for Exponential Random Variables,” Statistics and Probability Letters, 63, 361–365. DOI: 10.1016/S0167-7152(03)00101-9.
  • European Commission (2014), “Operation SNAKE: EU and Chinese Customs Join Forces to Target Undervaluation of Goods at Customs,” Press Release IP-14-1001, available at https://ec.europa.eu/commission/presscorner/detail/en/IP/_14/_1001.
  • Fernandez-Gracia, J., and Lacasa, L. (2018), “Bipartisanship Breakdown, Functional Networks, and Forensic Analysis in Spanish 2015 and 2016 National Elections,” Complexity, 2018, Article ID 9684749, DOI: 10.1155/2018/9684749.
  • Graham, R. L., Knuth, D. E., and Patashnik, O. (1994), Concrete Mathematics: A Foundation for Computer Science (2nd ed.), Reading, MA: Addison-Wesley.
  • Hill, T. P. (1995a), “The Significant-Digit Phenomenon,” The American Mathematical Monthly, 102, 322–327. DOI: 10.2307/2974952.
  • Hill, T. P. (1995b), “A Statistical Derivation of the Significant-Digit Law,” Statistical Science, 10, 354–363.
  • Hubert, M., Rousseeuw, P. J., and Van Aelst, S. (2008), “High-Breakdown Robust Multivariate Methods,” Statistical Science, 23, 92–119. DOI: 10.1214/088342307000000087.
  • Kossovsky, A. E. (2015), Benford’s Law: Theory, The General Law of Relative Quantities, and Forensic Fraud Detection Applications, Singapore: World Scientific.
  • Lacasa, L. (2019), “Newcomb-Benford Law Helps Customs Officers to Detect Fraud in International Trade,” Proceedings of the National Academy of Sciences of the United States of America, 116, 11–13. DOI: 10.1073/pnas.1819470116.
  • Luque, B., and Lacasa, L. (2009), “The First-Digit Frequencies of Prime Numbers and Riemann Zeta Zeros,” Proceedings of the Royal Society A, 465, 2197–2216. DOI: 10.1098/rspa.2009.0126.
  • Mebane, W. R., Jr. (2010), “Fraud in the 2009 Presidential Election in Iran?,” Chance, 23, 6–15. DOI: 10.1080/09332480.2010.10739785.
  • Mebane, W. R., Jr. (2011), “Comment on ‘Benford’s Law and the Detection of Election Fraud’,” Political Analysis, 19, 269–272.
  • Miller, S. J., ed. (2015), Benford’s Law: Theory and Applications, Princeton, NJ: Princeton University Press.
  • Miller, S. J., and Nigrini, M. (2008), “Order Statistics and Benford’s Law,” International Journal of Mathematics and Mathematical Sciences, 2008, Article ID 382948, DOI: 10.1155/2008/382948.
  • Nigrini, M. J. (1992), “The Detection of Income Tax Evasion Through an Analysis of Digital Distributions,” Ph.D. thesis, Department of Accounting, University of Cincinnati.
  • Nigrini, M. J. (2012), Benford’s Law, Hoboken, NJ: Wiley.
  • Pericchi, L., and Torres, D. (2011), “Quick Anomaly Detection by the Newcomb-Benford Law, With Applications to Electoral Processes Data From the USA, Puerto Rico and Venezuela,” Statistical Science, 26, 502–516. DOI: 10.1214/09-STS296.
  • Perrotta, D., Cerasa, A., Torti, F., and Riani, M. (2020), “The Robust Estimation of Monthly Prices of Goods Traded by the European Union,” Technical Report JRC120407, EUR 30188 EN, Publications Office of the European Union, Luxembourg. DOI: 10.2760/635844.
  • Pietronero, L., Tosatti, E., Tosatti, V., and Vespignani, A. (2001), “Explaining the Uneven Distribution of Numbers in Nature: The Laws of Benford and Zipf,” Physica A, 293, 297–304. DOI: 10.1016/S0378-4371(00)00633-6.
  • Raimi, R. A. (1976), “The First Digit Problem,” The American Mathematical Monthly, 83, 521–538. DOI: 10.1080/00029890.1976.11994162.
  • Riani, M., Atkinson, A. C., and Perrotta, D. (2014), “A Parametric Framework for the Comparison of Methods of Very Robust Regression,” Statistical Science, 29, 128–143. DOI: 10.1214/13-STS437.
  • Riani, M., Corbellini, A., and Atkinson, A. C. (2018), “The Use of Prior Information in Very Robust Regression for Fraud Detection,” International Statistical Review, 86, 205–218. DOI: 10.1111/insr.12247.
  • Rousseeuw, P., Perrotta, D., Riani, M., and Hubert, M. (2019), “Robust Monitoring of Time Series With Application to Fraud Detection,” Econometrics and Statistics, 9, 108–121. DOI: 10.1016/j.ecosta.2018.05.001.
  • Rubin-Delanchy, P., Heard, N. A., and Lawson, D. J. (2019), “Meta-Analysis of Mid-p-Values: Some New Results Based on the Convex Order,” Journal of the American Statistical Association, 114, 1105–1112. DOI: 10.1080/01621459.2018.1469994.
  • Serfling, R. J. (1980), Approximation Theorems of Mathematical Statistics, Hoboken, NJ: Wiley.
  • Tam Cho, W. K., and Gaines, B. J. (2007), “Breaking the (Benford) Law,” The American Statistician, 61, 218–223. DOI: 10.1198/000313007X223496.
  • Torti, F., Corbellini, A., and Atkinson, A. C. (2021), “fsdaSAS: a Package for Robust Regression for Very Large Datasets including the Batch Forward Search” (submitted).