test.KKurt {mnt} | R Documentation |
Computes the multivariate normality test based on the invariant measure of multivariate sample kurtosis due to Koziol (1989).
test.KKurt(data, MC.rep = 10000, alpha = 0.05)
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 |
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.
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.
Koziol, J.A. (1989), A note on measures of multivariate kurtosis, Biom. J., 31:619-624.
test.KKurt(MASS::mvrnorm(50,c(0,1),diag(1,2)),MC.rep=500)