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Thread: random question about non-Guassian multivariate distributions...

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    Re: random question about non-Guassian multivariate distributions...




    http://en.wikipedia.org/wiki/Elliptical_distribution

    The condition for a elliptical distribution is that there is a quadratic form inside the joint pdf. So I think only affine transformation can preserve this relationship.

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    Re: random question about non-Guassian multivariate distributions...

    Quote Originally Posted by BGM View Post
    So I think only affine transformation can preserve this relationship.
    so if i'm reading your message correctly your intuition would be that the resulting power-transformed RVs are not really part of the elliptical family of distributions... right?
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    Re: random question about non-Guassian multivariate distributions...

    Well a standard normal squared gives a chi-square which isn't in the elliptic family right? So I would say that you couldn't say that the power transformation is part of the elliptic family in general.
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    Re: random question about non-Guassian multivariate distributions...

    Quote Originally Posted by Dason View Post
    Well a standard normal squared gives a chi-square which isn't in the elliptic family right? So I would say that you couldn't say that the power transformation is part of the elliptic family in general.
    ta'da! and i got my answer.... AT LAST i mean, it relies on that claim about affine transfromation but upon thinking about it... it does seem like something reasonable....

    ... and all before the end of class!!! you people are awesome!
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    Re: random question about non-Guassian multivariate distributions...


    I would say that what is required is that any polynomial be of odd order (e.g. 3 or 5) where the transformation is strictly increasing - this implies that we would have a valid power method density. This also implies that the derivative is a polynomial of even order where the solutions to the zeros are all complex. For example, if we have polynomials of order 3 (i.e. Fleishman type), then the derivatives of the polynomials are quadratic and the associated discriminants must be negative --which has to be the case as the parametric form (real-two space) of a pdf for any marginal distribution is:

    f_{p\left ( z \right )}=\left ( p\left ( z \right ),\frac{\phi \left ( z \right )}{{p}'\left ( z \right )} \right )

    where you can see that the derivative of the polynomial is in the denominator of the ordinate and must be everywhere positive. Note that the numerator of the ordinate is the standard normal pdf.

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