mnt (version 1.3)

test.KKurt: Test of normality based on Koziols measure of multivariate sample kurtosis

Description

Computes the multivariate normality test based on the invariant measure of multivariate sample kurtosis due to Koziol (1989).

Usage

test.KKurt(data, MC.rep = 10000, alpha = 0.05)

Arguments

data

a n x d matrix of d dimensional data vectors.

MC.rep

number of repetitions for the Monte Carlo simulation of the critical value

alpha

level of significance of the test

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.

$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.

Details

Multivariate sample kurtosis due to Koziol (1989) is defined by $$\widetilde{b}_{n,d}^{(2)}=\frac{1}{n^2}\sum_{j,k=1}^n(Y_{n,j}^\top Y_{n,k})^4,$$ where \(Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n)\), \(j=1,\ldots,n\), 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. Note that for \(d=1\), we have a measure proportional to the squared sample kurtosis.

References

Koziol, J.A. (1989), A note on measures of multivariate kurtosis, Biom. J., 31:619-624.

See Also

KKurt

Examples

# NOT RUN {
test.KKurt(MASS::mvrnorm(50,c(0,1),diag(1,2)),MC.rep=500)

# }