usq.benftest {BenfordTests}

Freedman-Watson U-square Test for 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

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\left[ \displaystyle\sum_{i={10^{k-1}}}^{10^{k}-2}\left( \displaystyle\sum_{j=1}^{i}(f_j^o - f_j^e) \right)^2 - \frac{1}{9\cdot 10^{k-1}}\cdot\left(\displaystyle\sum_{i={10^{k-1}}}^{10^{k}-2}\displaystyle\sum_{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

Author(s)

Dieter William Joenssen [e-mail]

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.

See Also

pbenf, simulateH0

Examples

#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