test.BHEP {mnt}

Baringhaus-Henze-Epps-Pulley (BHEP) test

Description

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

Usage

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

Arguments

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

Details

The test statistic is

BHEP_{n,\beta}=\frac{1}{n} \sum_{j,k=1}^n \exp\left(-\frac{\beta^2\|Y_{n,j}-Y_{n,k}\|^2}{2}\right)- \frac{2}{(1+\beta^2)^{d/2}} \sum_{j=1}^n \exp\left(- \frac{\beta^2\|Y_{n,j}\|^2}{2(1+\beta^2)} \right) + \frac{n}{(1+2\beta^2)^{d/2}}.

Here, Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n), j=1,\ldots,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,\ldots,X_n. To ensure that the computation works properly n \ge d+1 is needed. If that is not the case the test returns an error.

Value

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.

#'

References

Henze, N., Wagner, T. (1997), A new approach to the class of BHEP tests for multivariate normality, J. Multiv. Anal., 62:1-23, DOI

See Also

BHEP

Examples

test.BHEP(MASS::mvrnorm(50,c(0,1),diag(1,2)),MC.rep=500)