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Benford and the Internal Capital Market: A Useful Indicator of Managerial Engagement Florian El Mouaaouy Ludwig-Maximilians-Universit€at M€unchen Jan Riepe Eberhard-Karls University T€ubingen Abstract. We propose the application of digit analysis using the Benford law to indi- cate managerial engagement in the capital allocation process. First, we motivate the potential of the Benford digit analysis to identify allocation outcomes that are shaped by human engagement instead of fixed decision rules. Second, we provide a case study to illustrate how the Benford

: Methods and Techniques for Forensic Accounting Investigations . John Wiley & Sons, New Jersey. 20. Omerzu, N., Kolar, I. (2019). Do the Financial Statements of Listed Companies on the Ljubljana Stock Exchange Pass the Benford's Law Test? International Business Research , Vol. 12, No. 1, pp. 54-64. 21. Papić, M., Vudrić, N., Jerin, K. (2017). Benfordov zakon i njegova primjena u forenzičkom računovodstvu. Zbornik sveučilišta Libertas, Vol. 1-2, No. 1-2, pp. 153-172. 22. Phillips, T. (2009). Simon Newcomb and “Natural Numbers” (Benford’s Law). American mathematical

data manipulation in general (IDW, 2006). If first digits of balance-sheet data or tax declarations deviate from Benford’s law, this evidence is taken by auditors and tax officers as a clue to check the records more carefully. In the United States, Nigrini (1992, 1996) was influential in establishing Benford’s law as an indicator of fraud in finance and taxation. He has shown that data in tax declarations follow the Benford law, whereas tax data known to be fraudulent do not. Such evidence induced tax authorities in the United States and Europe to check tax

(1998), ‘Using Benford’s Law and Neural Networks as a Review Procedure’, Managerial Auditing Journal 13, 356–366. Carslaw, C. A. P. (1988), ‘Anomalies in Income Numbers: Evidence of Goal Oriented Behaviour’, Accounting Review 63, 321–327. Cho, W. K. T. and B. J. Gaines (2007), ‘Breaking the (Benford) Law: Statistical Fraud Detection in Campaign Finance’, American Statistician 61, 218–223. Diekmann, A. (2007), ‘Not the First Digit! Using Benford’s Law to Detect Fraudulent Scientific Data’, Journal of Applied Statistics 34, 321–329. Diekmann, A. and B. Jann (2010

, $$\begin{array}{} \displaystyle x_{n} = A_{n}\, x_{n-1}, \quad n \in \mathbb{N}, \end{array} $$ where for every n , A n is a real m ṁ −-matrix, and where the problem is to study under what conditions the mantissa distribution generated for the trajectories with initial conditions x 0 ∈ ℝ d satisfy the Benford law. The results we obtain are related to the b - resonance condition introduced in [ 19 ]. Definition 6 A set Λ ⊂ ℂ is b −resonant if there exists a finite non-empty subset Λ 0 = { λ 1 , …, λ q ⊂ Λ with | λ 1 | = ⋅ s = | λ q | such that either card

sample C and the small sample D, which are considered as ‘clean’)and would then be detected quite quickly. We therefore focus our analysis on the first, second, and third waves of several SOEP subsamples. For testing the Benford Law procedures, we obtained (true) falsified records from the fieldwork organization that were previously detected using several conventional verifi- cation methods and statistical tests of stability and consistence (see Schräpler/Wagner 2005). Only one interviewer was able to fabricate data for the first two waves without raising suspicion

of occurrence in the case of the first two significant digits of a natural number to be [as reproduced in table 16.4]. BENFORD’S LAW 199 We can see that his first column heralded the Benford Law figures that we have derived: to establish the second we can proceed as follows. If we isolate the first two significant digits of a number by writing the number as x1x2 × 10n, where 10 x1x2 99, and define the random variable X accordingly, we have P(First significant digit is x1 and the second is x2) = P(x1x2 X < x1x2 + 1) = log10 ( 1+ 1 x1x2 ) . Now observe that P

/Kalinin 2 0 0 9 , Mebane 2010) . Besides Mebane's work, we can find further applications of Benford's Law-test to different political systems (e.g. Pericchi/Torres 2004 : Roukema 2009) . While there is an increasing number of application of this method, some scholars are skep- tical about the validity of the Benford's Law-test to detect fraud. According to Deckert, Myagkov and Ordeshook (2010), applying Benford's Law for detection of fraud lacks any theory or model. They further criticized that Benford's Law-tests can indicate electoral frauds in some innocent cases

Party Systems: Federalism and Party Competition in Canada, Great Britain, India, and the United States, Princeton University Press, Princeton, NJ, 2004. [ChWe] J. P. Chiverton and K. Wells, Mixture Effects in FIR Low-Pass Filtered Signals, IEEE Signal Processing Letters, 13(6), June 2006, 369–372. [ChGa] W. K. T. Cho and B. J. Gaines, Breaking the (Benford) Law: Statis- tical Fraud Detection in Campaign Finance, American Statistician 61 (2007), 218–223. [ChKTWZ] M. C. Chou, Q. X. Kong, C. P. Teo, Z. Z. Wang and H. Zheng, Ben- ford’s Law and Number Selection in Fixed

. Because the employees of the business have to obtain special approval for expenditures over $10,000, the fraudulent transactions are all for amounts between $9000 and $9999. For the 1000 legitimate expenditures, we have this data: 394 CHAPTER 20 First Digit Observed 1 314 2 178 3 111 4 92 5 88 6 59 7 56 8 56 9 46 Exercise 20.7.1. Using the Benford Law test at http://web.williams.edu/Mathematics/sjmiller/public html/benford /chapter01/MillerNigrini ExcelBenfordTester Ver401.xlsx (or any other suitable software), verify that the data conforms reasonably well to Benford