test.BHEP: Baringhaus-Henze-Epps-Pulley (BHEP) test
In mnt: Affine Invariant Tests of Multivariate Normality

Description Usage Arguments Details Value References See Also Examples

Performs the BHEP test of multivariate normality as suggested in Henze and Wagner (1997) using a tuning parameter a.

1	test.BHEP(data, a = 1, MC.rep = 10000, alpha = 0.05)

`data`	a n x d matrix of d dimensional data vectors.
`a`	positive numeric number (tuning parameter).
`MC.rep`	number of repetitions for the Monte Carlo simulation of the critical value
`alpha`	level of significance of the test

The test statistic is

BHEP_{n,β}=\frac{1}{n} ∑_{j,k=1}^n \exp≤ft(-\frac{β^2\|Y_{n,j}-Y_{n,k}\|^2}{2}\right)- \frac{2}{(1+β^2)^{d/2}} ∑_{j=1}^n \exp≤ft(- \frac{β^2\|Y_{n,j}\|^2}{2(1+β^2)} \right) + \frac{n}{(1+2β^2)^{d/2}}.

Here, Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n), j=1,…,n, are the scaled residuals, \overline{X}_n is the sample mean and S_n is the sample covariance matrix of the random vectors X_1,…,X_n. To ensure that the computation works properly n ≥ d+1 is needed. If that is not the case the test returns an error.

a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:

$Test: name of the test.
$param: value tuning parameter.
$Test.value: the value of the test statistic.
$cv: the approximated critical value.
$Decision: the comparison of the critical value and the value of the test statistic.