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 \(8d(d + 2)/n\).