mardiakurt: Multivariate Kurtosis
In jeksterslabds/jeksterslabRdist: Jekster's Lab Distribution Utilities

Description Usage Arguments Details Value Author(s) References Examples

Mardia's estimate of multivariate kurtosis is given by

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

where

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

1	mardiakurt(X)

`X`	Matrix or data frame.

If the null hypothesis that \mathbf{X} comes from a multivariate normal distribution is true, b_{2, k} follows a normal distribution with a mean of k ≤ft( k + 2 \right) and a variance of \frac{8 k ≤ft( k + 2 \right)}{n}. Consequently,

\frac{b_{2, k} - k ≤ft( k + 2 \right)}{√{\frac{8 k ≤ft( k + 2 \right)}{n}}}

asymptotically follows a standard normal distribution \mathcal{N} ≤ft( 0, 1 \right) .

Returns a vector with the following elements

b2: Estimate of multivariate kurtosis ≤ft( b_{2, k} \right) .
z: z-statistic ≤ft( \frac{b_{2, k} - k ≤ft( k + 2 \right)}{√{\frac{8 k ≤ft( k + 2 \right)}{n}}} \right) .
p: p-value associated with the z-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)
mardiakurt(X)