kurtosis

Mardia's multivariate skewness and kurtosis coefficients

Description

Functions to compute measures of multivariate skewness \((b_{1p})\) and kurtosis \((b_{2p})\) proposed by Mardia (1970),

\[b_{1p} = \frac{1}{n^2}\sum\limits_{i=1}^n\sum\limits_{j=1}^n ((\mathbf{x}_i -\overline{\mathbf{x}})^T\mathbf{S}^{-1}(\mathbf{x}_j - \overline{\mathbf{x}}))^3,\]

and

\[b_{2p} = \frac{1}{n}\sum\limits_{i=1}^n ((\mathbf{x}_i - \overline{\mathbf{x}})^T\mathbf{S}^{-1}(\mathbf{x}_j - \overline{\mathbf{x}}))^2.\]

Usage

kurtosis(x)
skewness(x)

Arguments

Argument	Description
x	matrix of data with, say, \(p\) columns.

References

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

Mardia, K.V., Zemroch, P.J. (1975). Algorithm AS 84: Measures of multivariate skewness and kurtosis. Applied Statistics 24 , 262-265.

Examples

setosa <- iris[1:50,1:4]
kurtosis(setosa)
skewness(setosa)