Mardia's estimate of multivariate kurtosis is given by $$ b_{2, k} = \frac{1}{n} \sum_{i = 1}^{n} \left[ \left( \mathbf{X}_{i} - \mathbf{\bar{X}} \right)^{T} \boldsymbol{\hat{\Sigma}}^{-1} \left( \mathbf{X}_{i} - \mathbf{\bar{X}} \right) \right]^{2} $$ 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
\(\boldsymbol{\hat{\Sigma}}\) is the estimated variance-covariance matrix of \(\mathbf{X}\) using sample data
mardiakurt(X)
| X | Matrix or data frame. |
|---|
Returns a vector with the following elements
Estimate of multivariate kurtosis \(\left( b_{2, k} \right)\) .
z-statistic \(\left( \frac{b_{2, k} - k \left( k + 2 \right)}{\sqrt{\frac{8 k \left( k + 2 \right)}{n}}} \right)\) .
p-value associated with the z-statistic.
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 \(k \left( k + 2 \right)\) and a variance of \(\frac{8 k \left( k + 2 \right)}{n}\). Consequently, $$ \frac{b_{2, k} - k \left( k + 2 \right)}{\sqrt{\frac{8 k \left( k + 2 \right)}{n}}} $$ asymptotically follows a standard normal distribution \(\mathcal{N} \left( 0, 1 \right)\) .
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)#> b2 b2.z b2.p #> 15.9079802 0.8288687 0.4071787