🚀 go-pugleaf

I earlier posted a message asking for help in finding names and/or
intuitive explanations for various central moments of bivariate
distributions.  In particular, what do mu(2,2), mu(2,1), and mu(1,2) tell
us about the bivariate distribution of X and Z, where 
mu(i,j) = E[(X-Xmean)^i (Z-Zmean)^j]?  The following is a summary of
useful responses I received to this question.

Ewart Shaw (strgh@uk.ac.warwick.cu) and Charles Berry
(cberry@tajo.ucsd.edu) independently suggested considering a contour
plot of the bivariate distribution.  Then, in Shaw's words,

>You could call mu(2,1), mu(1,2) & mu(2,2) `U-ness', `C-ness' & `X-ness'
>(positive values suggest that, `other things being equal', the contour
>plots of the distns. will have these shapes).

This suggestion was especially useful to me because in my problem
m(2,2) has a central role and it is clear in this problem why "X-ness"
would be important.

Terry Moore (T.Moore@massey.ac.nz) suggested:

>Look into Mardia's bivariate skewness and kurtosis. They use 
>mu(2,1), mu(1,2), and mu(2,2). You should be able to find these in
>the Encyclopaedia of Statistical Sciences.

Following this lead I found Mardia's 1970 Biometrika (Vol 57, 519-530)
paper to be very useful.  When X and Z are uncorrelated and have unit
variances, he shows that his definitions of bivariate skewness and
kurtosis reduce to:

bivariate skew = mu(3,0)^2 + mu(0,3)^2 + 3mu(1,2)^2 + 3mu(2,1)^2

bivariate kurtosis = mu(0,4) + mu(4,0) + 2mu(2,2)

I am most grateful for these useful suggestions for understanding
bivariate central moments.  Once again, I'm amazed by the power of the
electronic community.  If anyone has any further suggestions, I would
be eager to hear about them.

gary mcclelland
univ of colorado
mcclella@yertle.colorado.edu