test.MAKurt {mnt}

Test of normality based on multivariate kurtosis in the sense of Malkovich and Afifi

Description

Computes the multivariate normality test based on the invariant measure of multivariate sample kurtosis due to Malkovich and Afifi (1973).

Usage

test.MAKurt(data, MC.rep = 10000, alpha = 0.05, num.points = 1000)

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

num.points

number of points distributed uniformly over the sphere for approximation of the maximum on the sphere.

Details

Multivariate sample skewness due to Malkovich and Afifi (1973) is defined by

b_{n,d,M}^{(1)}=\max_{u\in \{x\in\mathbf{R}^d:\|x\|=1\}}\frac{\left(\frac{1}{n}\sum_{j=1}^n(u^\top X_j-u^\top \overline{X}_n )^3\right)^2}{(u^\top S_n u)^3},

where \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.

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.

$param

number of points used in approximation.

$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

Malkovich, J.F., and Afifi, A.A. (1973), On tests for multivariate normality, J. Amer. Statist. Ass., 68:176-179.

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

See Also

MAKurt

Examples

test.MAKurt(MASS::mvrnorm(10,c(0,1),diag(1,2)),MC.rep=100)