# jointdigit.benftest: A Hotelling _T-square_ Type Test for Benford's Law In BenfordTests: Statistical Tests for Evaluating Conformity to Benford's Law

## Description

`jointdigit.benftest` takes any numerical vector reduces the sample to the specified number of significant digits and performs a Hotelling T-square type goodness-of-fit test to assert if the data conforms to Benford's law.

## Usage

 ```1 2``` ```jointdigit.benftest(x = NULL, digits = 1, eigenvalues="all", tol = 1e-15, pvalmethod = "asymptotic", 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. `eigenvalues` How are the eigenvalues, which are used in testing, selected. `tol` Tolerance in detecting values that are essentially zero. `pvalmethod` Method used for calculating the p-value. Currently only `"asymptotic"` is available. `pvalsims` An integer specifying the number of replicates used if `pvalmethod = "simulate"`.

## Details

A Hotelling T^2 type goodness-of-fit test is performed on `signifd(x,digits)` versus `pbenf(digits)`. `x` is a numeric vector of arbitrary length. argument: `eigenvalues` can be defined as:

• numeric, a vector containing which eigenvalues should be used

• string length = 1, eigenvalue selection scheme:

• "all", use all non-zero eigenvalues

• "kaiser", use all eigenvalues larger than the mean of all non-zero eigenvalues

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 T^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 `eigenvalues_tested ` a vector containing the index numbers of the eigenvalues used in testing. `eigen_val_vect ` the eigen values and vectors of the null distribution. computed using `eigen`.

## References

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

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

`pbenf`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```#Set the random seed to an arbitrary number set.seed(421) #Create a sample satisfying Benford's law X<-rbenf(n=20) #Perform Test #on the sample's first digits using defaults jointdigit.benftest(X) #p-value = 0.648 #Perform Test #using only the two largest eigenvalues jointdigit.benftest(x=X,eigenvalues=1:2) #p-value = 0.5176 #Perform Test #using the kaiser selection criterion jointdigit.benftest(x=X,eigenvalues="kaiser") #p-value = 0.682 ```