MardiaMvtSkew: Mardia's Multivariate Skewness

In KbMvtSkew: Khattree-Bahuguna's Univariate and 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{μ} and positive definite variance-covariance matrix \boldsymbol{Σ}, Mardia's multivariate skewness is defined as

β_{1,p} = E[(\boldsymbol{X}_1 - \boldsymbol{μ})' \boldsymbol{Σ}^{-1} (\boldsymbol{X}_2 - \boldsymbol{μ})]^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, …, \boldsymbol{x}_n, its sample version is defined as

\hat{β}_{1,p} = \frac{1}{n^2} ∑_{i=1}^{n} ∑_{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}∑_{i=1}^{n} \boldsymbol{x}_i and the sample variance-covariance matrix \boldsymbol{S} = \frac{1}{n} ∑_{i=1}^{n} (\boldsymbol{x}_i - \bar{\boldsymbol{x}}) (\boldsymbol{x}_i - \bar{\boldsymbol{x}})'. It is assumed that n ≥ 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")])

KbMvtSkew documentation built on March 26, 2020, 7:44 p.m.