And you said that we were assuming a standard multivariate normal. Are you implying we work with no correlation between the Xs?

Dang! i should've said they are supposed to be correlated... sorry about that, yeah, they should be correlated.

Might be bit tedious to find the distribution of the transformed variables when dealing with polynomials. I assume this might result in non-standard distribution.

here's the line of reasoning of why i'm doing what i'm doing. 99.9% of simulation papers for non-normal, multivariate data rely on a paper published in Psychometrika. the full citation is:

Vale, C.D. and Maurelli, V.A. (1983) Simulating multivariate nonormal distributions, Psychometrika 48, 465-471.

the process is relatively straightforward and that is why people like it. starts with a) generate some multivarite normal vectors with a pre-specified correlation matrix. (b) do polynomial transofmations like i described and through the appropriate choice of constants which are multiplied by the powers of the random normal deviates you can end up generating random numbers with pre-specified correlations between them as well as pre-specified ammounts of skewness and kurtoses.

a guy by the name of Tadikamalla criticized this process precisely by saying something along the lines of "well, this is all very interesting but what do we know about the probability distribution of the random variable after it has gone through all these multiplyings and powerings?

now that i'm taking my seminar on non-Gaussian multivariate distributions we started talking about the relevance of moving away from from distributions whose contour (if we were to plot it) is elliptical. the prof proceeded to show us that there are a lot of distributions which can take all sorts of weird skewnesses and kurtoses (like the Pearson Type II distribution which looks like a cylinder) and still keep its elliptical shape.

this, of course brough back memories from the Vale & Maurelli paper which prompted me to ask something along the lines of

(a) let's take it back just a second and ask ourselves... not only do we know nothing about the distribution of such new variable after it is transformed through all these powers and whatnot, but do we even know if it is a distribution at all or just a whole bunch of numbers that behave the way we want?

and b...

(b) because these numbers were originated from a multivarite normal distribution with a set correlation matrix and, in the end, such correlations are still preserved even though we tansformed the variables to get all these crazy skewness and kurtoses.... could it be that the distribution we end up with is, in fact and ellpitical distribution so that it keeps its elliptical contour?

because if it does...well... in the light of what i've just learnt about how inadequate ellpitically-countoured distributions are to model some types of multivariate data... it could well be that the past 100 years of psychometric work on correlated, non-normal distributions needs some HUGE revisions because popular techniques such a structural equation modeling live and die by the assumption of normality...

psychometrics... NEEDS REVISION!! (could someone please cue in "Kaleidoscope of Mathematics" from my favourite movie.

*A Beautiful Mind*?