mardiaskew: Multivariate Skewness
In jeksterslabds/jeksterslabRdist: Jekster's Lab Distribution Utilities

Description Usage Arguments Details Value Author(s) References Examples

Mardia's estimate of multivariate skewness is given by

b_{1, k} = \frac{1}{n^2} ∑_{i = 1}^{n} ∑_{j = 1}^{n} ≤ft[ ≤ft( \mathbf{X}_{i} - \mathbf{\bar{X}} \right)^{T} \boldsymbol{\hat{Σ}}^{-1} ≤ft( \mathbf{X}_{j} - \mathbf{\bar{X}} \right) \right]^{3}

where

\mathbf{X} is the n \times k sample data
\mathbf{\bar{X}} represent sample means
\mathbf{X}_{i} - \mathbf{\bar{X}} and \mathbf{X}_{j} - \mathbf{\bar{X}} represent deviations from the mean
\boldsymbol{\hat{Σ}} is the estimated variance-covariance matrix of \mathbf{X} using sample data

1	mardiaskew(X)

`X`	Matrix or data frame.

If the null hypothesis that \mathbf{X} comes from a multivariate normal distribution is true, \frac{nb_{1, k}}{6} follows a chi-square ≤ft( χ^2 \right) distribution with a df of \frac{k(k + 1)(k + 2)}{6} .

Returns a vector with the following elements

b1: Estimate of multivariate skewness ≤ft( b_{1, k} \right) .
chisq: chi-square statistic ≤ft( \frac{nb_{1, k}}{6} \right) .
df: Degrees of freedom ≤ft( \frac{k(k + 1)(k + 2)}{6} \right) .
p: p-value associated with the chi-square statistic.

Ivan Jacob Agaloos Pesigan

Mardia, K. V. (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika, 57(3), 519-530. doi:10.2307/2334770.

Mardia, K. V. (1974). Applications of Some Measures of Multivariate Skewness and Kurtosis in Testing Normality and Robustness Studies. Sankhyā: The Indian Journal of Statistics, Series B (1960-2002), 36(2), 115-128.

set.seed(42)
n <- 100
mu <- c(0, 0, 0)
Sigma <- matrix(
  data = c(1, 0.5, 0.5, 0.5, 1, 0.5, 0.5, 0.5, 1),
  ncol = 3
)
X <- MASS::mvrnorm(n = n, mu = mu, Sigma = Sigma)
mardiaskew(X)