BHEP: Statistic of the BHEP-test

In mnt: Affine Invariant Tests of Multivariate Normality

Statistic of the BHEP-test

Description

This function returns the value of the statistic of the Baringhaus-Henze-Epps-Pulley (BHEP) test as in Henze and Wagner (1997).

Usage

BHEP(data, a = 1)

Arguments

data	a n x d matrix of d dimensional data vectors.
a	positive numeric number (tuning parameter).

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 function returns an error.

Value

value of the test statistic.

References

Henze, N., and Wagner, T. (1997), A new approach to the class of BHEP tests for multivariate normality, J. Multiv. Anal., 62:1–23, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1006/jmva.1997.1684")}

Epps T.W., Pulley L.B. (1983), A test for normality based on the empirical characteristic function, Biometrika, 70:723-726, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1093/biomet/70.3.723")}

Examples

BHEP(MASS::mvrnorm(50,c(0,1),diag(1,2)))

mnt documentation built on Sept. 25, 2026, 5:08 p.m.