KKurt {mnt}

Koziols measure of multivariate sample kurtosis

Description

This function computes the invariant measure of multivariate sample kurtosis due to Koziol (1989).

Usage

KKurt(data)

Arguments

data

a n x d matrix of d dimensional data vectors.

Details

Multivariate sample kurtosis due to Koziol (1989) is defined by

\widetilde{b}_{n,d}^{(2)}=\frac{1}{n^2}\sum_{j,k=1}^n(Y_{n,j}^\top Y_{n,k})^4,

where Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n), j=1,\ldots,n, are the scaled residuals, \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 kurtosis.

Value

value of sample kurtosis in the sense of Koziol.

References

Koziol, J.A. (1989), A note on measures of multivariate kurtosis, Biom. J., 31:619–624.

Examples

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