Description Details Author(s) References Examples

This package contains several specialized statistical tests and support functions for determining if numerical data could conform to Benford's law.

Package: | BenfordTests |

Type: | Package |

Version: | 1.2.0 |

Date: | 2015-07-18 |

License: | GPL-3 |

`BenfordTests`

is the implementation of eight goodness-of-fit (GOF) tests to assess if data conforms to Benford's law.

Tests include:

Pearson *χ^2* statistic (Pearson, 1900)

Kolmogorov-Smirnov *D* statistic (Kolmogorov, 1933)

Freedman's modification of Watson's *U^2* statistic (Freedman, 1981; Watson, 1961)

Chebyshev distance *m* statistic (Leemis, 2000)

Euclidean distance *d* statistic (Cho and Gaines, 2007)

Judge-Schechter mean deviation *a^** statistic (Judge and Schechter, 2009)

Joenssen's *J_P^2* statistic, a Shapiro-Francia type correlation test (Shapiro and Francia, 1972)

Joint Digit Test *T^2* statistic, a Hotelling type test (Hotelling, 1931)

All tests may be performed using more than one leading digit.
All tests simulate the specific p-values required for statistical inference, while p-values for the *χ^2*, *D*, *a^**, and *T^2* statistics may also be determined using their asymptotic distributions.

Dieter William Joenssen

Maintainer: Dieter William Joenssen <Dieter.Joenssen@googlemail.com>

Benford, F. (1938) The Law of Anomalous Numbers. *Proceedings of the American Philosophical Society*. **78**, 551–572.

Cho, W.K.T. and Gaines, B.J. (2007) Breaking the (Benford) Law: Statistical Fraud Detection in Campaign Finance. *The American Statistician*. **61**, 218–223.

Freedman, L.S. (1981) Watson's Un2 Statistic for a Discrete Distribution. *Biometrika*. **68**, 708–711.

Joenssen, D.W. (2013) Two Digit Testing for Benford's Law. *Proceedings of the ISI World Statistics Congress, 59th Session in Hong Kong*. [available under http://www.statistics.gov.hk/wsc/CPS021-P2-S.pdf]

Judge, G. and Schechter, L. (2009) Detecting Problems in Survey Data using Benford's Law. *Journal of Human Resources*. **44**, 1–24.

Kolmogorov, A.N. (1933) Sulla determinazione empirica di una legge di distibuzione. *Giornale dell'Istituto Italiano degli Attuari*. **4**, 83–91.

Leemis, L.M., Schmeiser, B.W. and Evans, D.L. (2000) Survival Distributions Satisfying Benford's law. *The American Statistician*. **54**, 236–241.

Newcomb, S. (1881) Note on the Frequency of Use of the Different Digits in Natural Numbers. *American Journal of Mathematics*. **4**, 39–40.

Pearson, K. (1900) On the Criterion that a Given System of Deviations from the Probable in the Case of a Correlated System of Variables is Such that it can be Reasonably Supposed to have Arisen from Random Sampling. *Philosophical Magazine Series 5*. **50**, 157–175.

Shapiro, S.S. and Francia, R.S. (1972) An Approximate Analysis of Variance Test for Normality. *Journal of the American Statistical Association*. **67**, 215–216.

Watson, G.S. (1961) Goodness-of-Fit Tests on a Circle. *Biometrika*. **48**, 109–114.

Hotelling, H. (1931). The generalization of Student's ratio. *Annals of Mathematical Statistics*. **2**, 360–378.

1 2 3 4 5 6 7 8 9 10 11 12 | ```
#Set the random seed to an arbitrary number
set.seed(421)
#Create a sample satisfying Benford's law
X<-rbenf(n=20)
#Look at sample
X
#Look at the first digits of the sample
signifd(X)
#Perform a Chi-squared Test on the sample's first digits using defaults
chisq.benftest(X)
#p-value = 0.648
``` |

```
[1] 6.159420 1.396476 5.193371 2.064033 7.001284 5.006184 7.950332 4.822725
[9] 3.386809 1.619609 2.080063 2.242473 1.944697 5.460581 6.443031 2.662821
[17] 2.079283 3.703353 1.364175 3.354136
[1] 6 1 5 2 7 5 7 4 3 1 2 2 1 5 6 2 2 3 1 3
Chi-Square Test for Benford Distribution
data: X
chisq = 5.9932, p-value = 0.648
```

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