This function computes the invariant measure of multivariate sample skewness due to Móri, Rohatgi and Székely (1993).
MRSSkew(data)value of sample skewness in the sense of Móri, Rohatgi and Székely.
a n x d matrix of d dimensional data vectors.
Multivariate sample skewness due to Móri, Rohatgi and Székely (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 function returns an error. Note that for \(d=1\), it is equivalent to skewness in the sense of Mardia.
Móri, T. F., Rohatgi, V. K., Székely, 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.