fastmatrix (version 0.5-7721)

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), $$b_{1p} = \frac{1}{n^2}\sum\limits_{i=1}^n\sum\limits_{j=1}^n ((\bold{x}_i - \overline{\bold{x}})^T\bold{S}^{-1}(\bold{x}_j - \overline{\bold{x}}))^3,$$ and $$b_{2p} = \frac{1}{n}\sum\limits_{i=1}^n ((\bold{x}_i - \overline{\bold{x}})^T \bold{S}^{-1}(\bold{x}_j - \overline{\bold{x}}))^2.$$

Usage

kurtosis(x)
skewness(x)

Arguments

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)