test.MRSSkew {mnt}

Test of multivariate normality based on the measure of multivariate skewness of Mori, Rohatgi and Szekely

Description

Computes the multivariate normality test based on the invariant measure of multivariate sample skewness due to Mori, Rohatgi and Szekely (1993).

Usage

test.MRSSkew(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

Details

Multivariate sample skewness due to Mori, Rohatgi and Szekely (1993) is defined by

\widetilde{b}_{n,d}^{(1)}=\frac{1}{n}\sum_{j=1}^n\|Y_{n,j}\|^2\|Y_{n,k}\|^2Y_{n,j}^\top Y_{n,k},

where Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n), \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, it is equivalent to skewness in the sense of Mardia.

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.

References

Mori, T. F., Rohatgi, V. K., Szekely, G. J. (1993), On multivariate skewness and kurtosis, Theory of Probability and its Applications, 38:547-551.

Henze, N. (2002), Invariant tests for multivariate normality: a critical review, Statistical Papers, 43:467-506.

See Also

MRSSkew

Examples

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