Finding Mardia's multivariate kurtosis of multiple variables
Usage
mardiaKurtosis(dat)
Arguments
dat
The target matrix or data frame with multiple variables
Value
A value of a Mardia's multivariate kurtosis with a test statistic
Details
The Mardia's multivariate kurtosis formula (Mardia, 1970) is
$$b_{2, d} = \frac{1}{n}\sum^n_{i=1}\left[ \left(\bold{X}_i - \bold{\bar{X}} \right)^{'} \bold{S}^{-1} \left(\bold{X}_i - \bold{\bar{X}} \right) \right]^2,$$
where $d$ is the number of variables, $X$ is the target dataset with multiple variables, $n$ is the sample size, $\bold{S}$ is the sample covariance matrix of the target dataset, and $\bold{\bar{X}}$ is the mean vectors of the target dataset binded in $n$ rows. When the population multivariate kurtosis is normal, the $b_{2,d}$ is asymptotically distributed as normal distribution with the mean of $d(d + 2)$ and variance of $8d(d + 2)/n$.
References
Mardia, K. V. (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika, 57, 519-530.