test.BHEP: Baringhaus-Henze-Epps-Pulley (BHEP) test

In mnt: Affine Invariant Tests of Multivariate Normality

Baringhaus-Henze-Epps-Pulley (BHEP) test

Description

Performs the BHEP test of multivariate normality of Henze and Wagner (1997), with tuning parameter a. Critical values and p-values are obtained by Monte Carlo simulation or by fitting a Pearson or Johnson distribution to the first four moments of the limiting null distribution.

Usage

test.BHEP(
  data,
  a = 1,
  MC.rep = 10000,
  alpha = 0.05,
  cv.method = c("MC", "Pearson", "Johnson")
)

Arguments

data	A numeric n \times d matrix with observations in rows. A numeric vector is treated as a univariate sample. All entries must be finite, n \ge d+1, and the sample covariance must be nonsingular.
a	Positive, finite tuning parameter; corresponds to \beta in the formulas below.
MC.rep	Positive integer giving the number of Monte Carlo replications. Used only when cv.method = "MC".
alpha	Significance level, strictly between zero and one.
cv.method	Method used to approximate the critical value: "MC" (default), "Pearson", or "Johnson". Partial matching is supported.

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 \beta=a, Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n), and S_n=n^{-1}\sum_{j=1}^n (X_j-\overline{X}_n)(X_j-\overline{X}_n)^\top.

With cv.method = "MC", cv.quan simulates the null distribution at the observed sample size n and dimension d, using MC.rep replications. The same simulated statistics are used for the critical value and the Monte Carlo p-value

p_{MC}=\frac{1+\sum_{b=1}^{B}\mathbf{1}\{T_b\ge T_{obs}\}}{B+1},

where B is MC.rep. The critical value retains the order-statistic convention of cv.quan, and Decision remains Test.value > cv. Because the Monte Carlo p-value is discrete and uses the plus-one correction, this decision can differ from pvalue <= alpha near the cutoff.

With cv.method = "Pearson" or "Johnson", the critical value is obtained from a moment-based approximation to the limiting null distribution T_\beta(d). Let \kappa_r denote its cumulants. The first three cumulants are given by Henze and Wagner (1997), and the fourth by Theorem 3.2 and Appendix A of Ebner and Henze's announced manuscript. The mean, variance, skewness and ordinary kurtosis are

\mu=\kappa_1,\quad \sigma^2=\kappa_2,\quad \gamma_1=\kappa_3/\kappa_2^{3/2},\quad \beta_2=3+\kappa_4/\kappa_2^2.

In particular, the fourth central moment is \mu_4=\kappa_4+3\kappa_2^2; it is not the fourth cumulant.

Value

An object of class "mnt", with components:

Test

Name of the test.

param

Value of the tuning parameter.

Test.value

Value of the test statistic.

cv

Approximated upper 1-\alpha critical value.

Decision

TRUE if Test.value > cv, so the null hypothesis is rejected at the chosen significance level.

cv.method

Critical-value method actually used.

pvalue

Upper-tail p-value computed using the selected method. The Pearson and Johnson p-values are asymptotic approximations.

References

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

Ebner, B. and Henze, N. Four-Moment Approximations and Fast p-Values for BHEP Tests of Multivariate Normality. Supplied manuscript ‘BHEP_fourth_moment.pdf’, Theorem 3.2 and Appendix A.

See Also

BHEP, cv.quan

Examples

set.seed(123)
x <- matrix(rnorm(100), ncol = 2)
test.BHEP(x, MC.rep = 500)
if (requireNamespace("PearsonDS", quietly = TRUE)) {
  test.BHEP(x, cv.method = "Pearson")
}
if (requireNamespace("SuppDists", quietly = TRUE)) {
  test.BHEP(x, cv.method = "Johnson")
}

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