MSkew {mnt}

Mardias measure of multivariate sample skewness

Description

This function computes the classical invariant measure of multivariate sample skewness due to Mardia (1970).

Usage

MSkew(data)

Arguments

data

a n x d matrix of d dimensional data vectors.

Details

Multivariate sample skewness due to Mardia (1970) is defined by

b_{n,d}^{(1)}=\frac{1}{n^2}\sum_{j,k=1}^n(Y_{n,j}^\top Y_{n,k})^3,

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, we have a measure proportional to the squared sample skewness.

Value

value of sample skewness in the sense of Mardia.

References

Mardia, K.V. (1970), Measures of multivariate skewness and kurtosis with applications, Biometrika, 57:519–530.

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

Examples

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