Why aren't the coskewness and cokurtosis matrices square like the covariance matrix?

Question

The variance-covariance matrix is shaped $p\times p$, whereas the co-skewness matrix is shaped $p\times p^2$ and the co-kurtosis matrix is $p\times p^3$. Why is this, given that skewness and kurtosis are merely the 3rd and 4th moments after variance, the 2nd moment? Is there something fundamentally much different in the formula of skewness and kurtosis than variance that makes their matrices much larger and non-square?

Answer 1

The statistics aren't really $p\times p^2$ and $p\times p^3$ matrices. They are really $p\times p \times p$ and $p\times p \times p\times p$ arrays (covariant tensors of rank 3 and 4), but because we like representing things in two dimensions they get folded into non-square matrices.

The coskewness is $p\times p \times p$ because it has $(i,j,k)$ value $E[X_iX_jX_k]$; similarly, the co-kurtosis represents expectations of products of four variables and so is a 4-way array.