MAKurt {mnt}

multivariate kurtosis in the sense of Malkovich and Afifi

Description

This function computes the invariant measure of multivariate sample kurtosis due to Malkovich and Afifi (1973).

Usage

MAKurt(data, Points = NULL)

Arguments

data

a n x d matrix of d dimensional data vectors.

Points

points for approximation of the maximum on the sphere. Points=NULL generates 1000 uniformly distributed Points on the d dimensional unit 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 function returns an error.

Value

value of sample kurtosis in the sense of Malkovich and Afifi.

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.

Examples

MAKurt(MASS::mvrnorm(50,c(0,1),diag(1,2)))