MASkew: multivariate skewness in the sense of Malkovich and Afifi
Description
This function computes the invariant measure of multivariate sample skewness due to Malkovich and Afifi (1973).
Usage
MASkew(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.
Value
value of sample skewness in the sense of Malkovich and Afifi.
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.
References
Malkovich, J.F., and Afifi, A.A. (1973), On tests for multivariate normality, J. Amer. Statist. Ass., 68:176<U+2013>179.
Henze, N. (2002), Invariant tests for multivariate normality: a critical review, Statistical Papers, 43:467<U+2013>506.