Is Y_i always a function of X_i alone or could it be a function of more than one of the X_js?
today in my seminar we were talking about all these interesting ways people can start with random uniform distributions [0,1] and through doing and re-doing things to them end up with pretty much any well-known probability distribution...
.... i was wondering whether something along those lines is true of any distribution (or does it even keep its properties at all).
for the very specific example i have in mind.... say you have X1, X2, X3..Xn which all follow a multivariate normal distribution (say a standard one so we get our nice vector of 0's for a mean and a correlation matrix).
if i start creating new variables say Y1 = aX1 + bX1^2 +cX1^3, Y2=dX2 + eX2^2+ fX2^3 and so on... (so, in other words, i am creating new variables Yi which are the result of some polynomial expressions of my original multivariate standard normal Xi's) ...
... is there a way to know something about the distribution of <Y1, Y2, Y3,...,Yn>? or could it be that those Ys are not even a probability distribution anymore? any insight into how to get the density/cdf?
for all your psychometric needs! https://psychometroscar.wordpress.com/about/
Is Y_i always a function of X_i alone or could it be a function of more than one of the X_js?
I don't have emotions and sometimes that makes me very sad.
for simplicity let's start assuming it's just a function of it's corresponding Xi alone and let's see where it goes... lol.
for all your psychometric needs! https://psychometroscar.wordpress.com/about/
And you said that we were assuming a standard multivariate normal. Are you implying we work with no correlation between the Xs? In that case it's just a matter of finding the distribution of a polynomial transformation?
I don't have emotions and sometimes that makes me very sad.
spunky (03-09-2012)
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.
Oh Thou Perelman! Poincare's was for you and Riemann's is for me.
spunky (03-09-2012)
Dang! i should've said they are supposed to be correlated... sorry about that, yeah, they should be correlated.
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?
for all your psychometric needs! https://psychometroscar.wordpress.com/about/
Spunky, I solved this problem that Tadikamalla raised in his 1980 article. In short, the pdf and cdf for power method polynomials has to be expressed in parametric form (real-two space). The first article I wrote on this was a JSCS article published in 2007.
You can also see the derivation of the pdf and cdf in my book on pages 9-14.
for all your psychometric needs! https://psychometroscar.wordpress.com/about/
I think it depends on what it is a researcher is doing...Are you referring to higher order moments? or other things?
i guess what i am asking more precisely would be:
for the transfromed variable (as discussed above with all the multiplying and powering), in the d > 2 space...
(a) does it have a legit density function (so that it's non-negative and integrates to 1)?
(b) does it belong to the family of elliptically-contoured distributions? (which i guess could be answered from a if there is a closed-form expression)
for all your psychometric needs! https://psychometroscar.wordpress.com/about/
I would say yes, so long as the transformations are strictly increasing. I good place to start on this topic would be to look at Chapter 9 in Karian & Dudewicz (2011) in the context of the Generalized Lambda Distribution (see pages 363-414) because the idea is similar to what you're asking. They develop a nice extension of a bivariate GLD using Plackett's Method of a bivariate cdf construction that has to consider the Fretchet upper and lower bounds. I'm relatively certain that this approach could be used in the context of power method polynomials....Sounds like a good toipic for a dissertation. :-)
References:
Karian, Z. A., and Dudewicz, E. J. (2011) Handbook of Fitting Statistical Distributions with R. Chapman & Hall/CRC, Boca Raton, FL.
Plackett, R. L. (1965). A class of bivariate distributions. Biometrica, 60, 516-562.
spunky (03-09-2012)
so.... is that a yes to (a) and (b) implying that it has a legit pdf and resulting power-transformed RVs belong to the family of elliptical distributions... or just a yes to the (a)... or to the (b) parts?
oh thank you! THIS is what i'm after so i dont have to spend hours and hours on the internet trying to see whether this is even feasible or not...
i'll keep it in the bucket list then for the PhD... gotta get that MA thesis done first but it's always nice to start looking at topics early on...
Thank you very much Dragan, ledzep and Dason. i think i shall proceed to spread luv & thanks to everyone...
for all your psychometric needs! https://psychometroscar.wordpress.com/about/
Actually I wonder why Spunky have doubts on the transformation part.
As long as the original random variable is absolutely continuous, and the function has countable many critical points only, (this is the condition I guess)
then the resulting transformed variable should be again absolutely continuous and with a legit pdf.
Of course in multivariate case it is provided that the random variables does not have any functional relationship: e.g. if X = -Y, then Z = X + Y is identical to zero.
I am not sure about the condition to preserve an elliptic family after the transformation.
spunky (03-09-2012)
thanks BGM! i guess because i haven't seen the actual pdf for the resulting power-transformed RV (in the case of greater than 2 dimensions) i had trouble seeing it... but you're right, i didnt think about it from this perspective. thanks
this is the part that i'd REALLY want to know...
for all your psychometric needs! https://psychometroscar.wordpress.com/about/
Tweet |