usq.benftest: Freedman-Watson U-square Test for Benford's Law In BenfordTests: Statistical Tests for Evaluating Conformity to Benford's Law

Description

usq.benftest takes any numerical vector reduces the sample to the specified number of significant digits and performs the Freedman-Watson test for discreet distributions between the first digits' distribution and Benford's distribution to assert if the data conforms to Benford's law.

Usage

 1 usq.benftest(x = NULL, digits = 1, pvalmethod = "simulate", pvalsims = 10000) 

Arguments

 x A numeric vector. digits An integer determining the number of first digits to use for testing, i.e. 1 for only the first, 2 for the first two etc. pvalmethod Method used for calculating the p-value. Currently only "simulate" is available. pvalsims An integer specifying the number of replicates used if pvalmethod = "simulate".

Details

A Freedman-Watson test for discreet distributions is performed between signifd(x,digits) and pbenf(digits). Specifically:

U^2 = \frac{n}{9\cdot 10^{k-1}}\cdot≤ft[ \displaystyle∑_{i={10^{k-1}}}^{10^{k}-2}≤ft( \displaystyle∑_{j=1}^{i}(f_j^o - f_j^e) \right)^2 - \frac{1}{9\cdot 10^{k-1}}\cdot≤ft(\displaystyle∑_{i={10^{k-1}}}^{10^{k}-2}\displaystyle∑_{j=1}^{i}(f_i^o - f_i^e)\right)^2\right]

where f_i^o denotes the observed frequency of digits i, and f_i^e denotes the expected frequency of digits i. x is a numeric vector of arbitrary length. Values of x should be continuous, as dictated by theory, but may also be integers. digits should be chosen so that signifd(x,digits) is not influenced by previous rounding.

Value

A list with class "htest" containing the following components:

 statistic  the value of the U^2 test statistic p.value  the p-value for the test method  a character string indicating the type of test performed data.name  a character string giving the name of the data

References

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

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]

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

pbenf, simulateH0
 1 2 3 4 5 6 7 8 #Set the random seed to an arbitrary number set.seed(421) #Create a sample satisfying Benford's law X<-rbenf(n=20) #Perform Freedman-Watson U-squared Test on #the sample's first digits using defaults usq.benftest(X) #p-value = 0.4847