Visualizing higher-order cross-moments (cokurtosis, coskewness)

Question

How could and coskewness and cokurtosis be visualized in an easily comprehensible manner?

Mean, variances, skewness, kurtosis can easily be illustrated in density plots:

(Source: own *TeX-stuff)

The first cross-moment (co-variance) can be visualized by looking at joint densities:

(Source: https://www.quora.com/Is-it-possible-to-visualize-covariance-in-a-bivariate-normal-distribution-in-a-straightforward-way-similar-to-variance-in-a-univariate-normal-distribution)

How could and coskewness and cokurtosis be visualized in an easily comprehensible manner?

Whereby with coskewness (between two random variables) I mean $$\frac{\operatorname{E} \left[(X - \operatorname{E}[X])^2(Y - \operatorname{E}[Y])\right]}{\sigma_X^2 \sigma_Y}.$$ And with cokurtosis I mean both, the asymmetric $$E= {\operatorname{E}{\big[(X - \operatorname{E}[X])^3(Y - \operatorname{E}[Y])\big]} \over \sigma_X^3 \sigma_Y}$$ as well as the symetric cokurtosis $${\operatorname{E}{\big[(X - \operatorname{E}[X])^2(Y - \operatorname{E}[Y])^2\big]} \over \sigma_X^2 \sigma_Y^2}.$$

The purpose of this question is to be able to better explain these concepts to beginners

Answer 1

My case maybe a little different, recently I am running into a similar problem. I am trying to visualize a positive triple coskewness of 3 zero mean random variables, written as $\langle X_1X_2X_3\rangle$.

From the case of positive covariance, we have $\langle X_1X_2\rangle$. The computation of covariance itself does not distinguish between the cases of both $X_1$, $X_2$ are positive, or both of them are negative. As a result, if we plot the $X_1$ against $X_2$ in two dimensional space, we tend to see a line with slope = 1, as the distribution has high occurrence when both are positive or both are negative.

Now going back to the triple coskewness, there are four such degeneracies: $\langle X_1X_2X_3\rangle$, $\langle X_1-X_2-X_3\rangle$, $\langle -X_1-X_2X_3\rangle$, $\langle -X_1X_2-X_3\rangle$ It turns out, if we plot them in a three-dimensional space, with the value of $X_1$, $X_2$ and $X_3$ as the XYZ-axis, we should get a tetrahedron profile for this distribution.

Here is what it looks like when I tried to plot such distribution (with artificially added noise to simulate imperfect coskewness):

Please note that this is merely my guess, I don't have much supporting as to whether the above is true. I am also looking for references on it, great if you happen to have anything to share.