Editing Benford's law (section)

==Statistical tests==
Although the [[chi-squared test]] has been used to test for compliance with Benford's law it has low statistical power when used with small samples.

The [[Kolmogorov–Smirnov test]] and the [[Kuiper test]] are more powerful when the sample size is small, particularly when Stephens's corrective factor is used.<ref name=Stephens1970>{{cite journal |last=Stephens |first=M. A. |year=1970 |title=Use of the Kolmogorov–Smirnov, Cramér–von Mises and related statistics without extensive tables |journal=[[Journal of the Royal Statistical Society, Series B]] |volume=32 |issue=1 |pages=115–122 |doi=10.1111/j.2517-6161.1970.tb00821.x }}</ref> These tests may be unduly conservative when applied to discrete distributions. Values for the Benford test have been generated by Morrow.<ref name=Morrow2014>{{cite book |last = Morrow |first = John |title = Benford's Law, families of distributions and a test basis |location = London, UK |access-date = 2022-03-11 |date = August 2014 |url = http://cep.lse.ac.uk/_new/publications/series.asp?prog=CEP}}</ref> The critical values of the test statistics are shown below:

:{| class="wikitable" style="text-align:center;"
! {{diagonal split header|Test| {{mvar|⍺}} }}
! 0.10
! 0.05
! 0.01
|-
| Kuiper
| 1.191
| 1.321
| 1.579
|-
| Kolmogorov–Smirnov
| 1.012
| 1.148
| 1.420
|}

These critical values provide the minimum test statistic values required to reject the hypothesis of compliance with Benford's law at the given [[significance level]]s.

Two alternative tests specific to this law have been published: First, the max ({{mvar|m}}) statistic<ref name=Leemis2000>{{cite journal |last1=Leemis |first1=L. M. |last2=Schmeiser |first2=B. W. |last3=Evans |first3=D. L. |year=2000 |title=Survival distributions satisfying Benford's Law |journal=The American Statistician |volume=54 |issue=4 |pages=236–241 |doi=10.1080/00031305.2000.10474554 |s2cid=122607770}}</ref> is given by

: <math>m = \sqrt{N} \cdot \max_{k=1}^9 \left\{\left|\Pr\left(X \text{ has FSD} = k\right) - \log_{10}\left(1 + \frac{1}{k}\right)\right|\right\}.</math>

The leading factor <math>\sqrt{N}</math> does not appear in the original formula by Leemis;<ref name=Leemis2000/> it was added by Morrow in a later paper.<ref name=Morrow2014/>

Secondly, the distance ({{mvar|d}}) statistic<ref name=Cho2007>{{cite journal |last1=Cho |first1=W. K. T. |last2=Gaines |first2=B. J. |year=2007 |title=Breaking the (Benford) law: Statistical fraud detection in campaign finance |journal=The American Statistician |volume=61 |issue=3 |pages=218–223 |doi=10.1198/000313007X223496 |s2cid=7938920 }}</ref> is given by

: <math>d = \sqrt{N \cdot \sum_{l=1}^9 \left[\Pr\left(X \text{ has FSD} = l\right) - \log_{10}\left(1 + \frac{1}{l}\right)\right]^2},</math>

where FSD is the first significant digit and {{mvar|N}} is the sample size. Morrow has determined the critical values for both these statistics, which are shown below:<ref name=Morrow2014/>

:{| class="wikitable" style="text-align:center;"
! {{diagonal split header|Statistic|{{mvar|⍺}}}}
! 0.10
! 0.05
! 0.01
|-
| Leemis's {{mvar|m}} 
| 0.851 
| 0.967 
| 1.212 
|-
| Cho & Gaines's {{mvar|d}} 
| 1.212
| 1.330
| 1.569
|}

Morrow has also shown that for any random variable  {{mvar|X}}  (with a continuous [[probability density function|PDF]]) divided by its standard deviation ({{mvar|σ}}), some value  {{mvar|A}}  can be found so that the probability of the distribution of the first significant digit of the random variable <math>|X/\sigma|^A</math> will differ from Benford's law by less than {{nobr|{{mvar|ε}} > 0.}}<ref name=Morrow2014/> The value of {{mvar|A}} depends on the value of {{mvar|ε}} and the distribution of the random variable.

A method of accounting fraud detection based on bootstrapping and regression has been proposed.<ref name=Suh2011>{{cite journal |last1=Suh |first1=I. S. |last2=Headrick |first2=T. C. |last3=Minaburo |first3=S. |year=2011 |title=An effective and efficient analytic technique: A bootstrap regression procedure and Benford's Law |journal=J. Forensic & Investigative Accounting |volume=3 |issue=3}}</ref>

If the goal is to conclude agreement with the Benford's law rather than disagreement, then the [[goodness-of-fit test]]s mentioned above are inappropriate. In this case the specific [[Equivalence test|tests for equivalence]] should be applied. An empirical distribution is called equivalent to the Benford's law if a distance (for example total variation distance or the usual Euclidean distance) between the probability mass functions is sufficiently small. This method of testing with application to Benford's law is described in Ostrovski.<ref>{{cite journal |last=Ostrovski |first=Vladimir |date=May 2017 |title=Testing equivalence of multinomial distributions |journal=Statistics & Probability Letters |volume=124 |pages=77–82 |doi=10.1016/j.spl.2017.01.004 |s2cid=126293429 |url=https://www.researchgate.net/publication/312481284 }}</ref>