| MardiaMvtSkew {KbMvtSkew} | R Documentation |
Mardia's Multivariate Skewness
Description
Compute Mardia's Multivariate Skewness.
Usage
MardiaMvtSkew(x)
Arguments
x |
a matrix of original observations. |
Details
Given a p-dimensional multivariate random vector with mean vector \boldsymbol{\mu} and positive definite variance-covariance matrix \boldsymbol{\Sigma}, Mardia's multivariate skewness is defined as
\beta_{1,p} = E[(\boldsymbol{X}_1 - \boldsymbol{\mu})' \boldsymbol{\Sigma}^{-1} (\boldsymbol{X}_2 - \boldsymbol{\mu})]^3,
where \boldsymbol{X}_1 and \boldsymbol{X}_2 are independently and identically distributed copies of \boldsymbol{X}. For a multivariate random sample of size n, \boldsymbol{x}_1, \boldsymbol{x}_1, \ldots, \boldsymbol{x}_n, its sample version is defined as
\hat{\beta}_{1,p} = \frac{1}{n^2} \sum_{i=1}^{n} \sum_{j=1}^{n} [(\boldsymbol{x}_i - \bar{\boldsymbol{x}})'\boldsymbol{S}^{-1} (\boldsymbol{x}_j - \bar{\boldsymbol{x}})]^3,
where the sample mean \bar{\boldsymbol{x}} = \frac{1}{n}\sum_{i=1}^{n} \boldsymbol{x}_i and the sample variance-covariance matrix \boldsymbol{S} = \frac{1}{n} \sum_{i=1}^{n} (\boldsymbol{x}_i - \bar{\boldsymbol{x}}) (\boldsymbol{x}_i - \bar{\boldsymbol{x}})'. It is assumed that n \ge p.
Value
MardiaMvtSkew gives the sample Mardia's multivairate skewness.
References
Mardia, K.V. (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika, 57(3), 519–530.
Examples
# Compute Mardia's multivairate skewness
data(OlymWomen)
MardiaMvtSkew(OlymWomen[, c("m800","m1500","m3000","marathon")])