From: John Fowler
4 Attachments, 6 KB
Frank,
Believe it or not, the cokurtosis turns out to play a role in the variance of the "erroneous" chi- square. Actually, it's just the fourth moments x^3y and xy^3. But as a related concept, it raises the question we didn't address before: how do you normalize coskewness and cokurtosis, i.e., the usual kurtosis is the fourth central moment divided by sigma^4; for x^3y, I assume you would divide by sigma(x)^3*sigma(y). In that case, this particular cokurtosis (of the five possible) is 3*rho. So for uncorrelated variables, this cokurtosis is zero. But I'm concerned with correlated variables, the 2-D Gaussians.
For correlated zero-mean Gaussian x and y, the correct and erroneous chi-squares are:
The difference is
The expectation value of this difference is zero, as we have seen previously. The expected squared value of the difference is:
You can see that when you expand the square, you're going to pick up terms in x^3y and xy^3, so the expectation values of these will be needed. You can't just factor these and use the expectation values of the factors, e.g.,
There you can see that if you divide the last result by sigma(x)^3*sigma(y), you get 3*rho for the cokurtosis, zero if rho is zero.
I had to coax Maple a bit to get that definite double integral out of it; the key is declaring all variables as real, the sigmas > 0, and -1 < rho < 1. Until I did that, it was spewing all kinds of conditional results, including divergences. So while it's fresh in my mind, I plan to compute all the other cokurtoses.
I thought you'd enjoy finding out that an actual use for a cokurtosis-related concept arose in practice!
Regards, John