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.
for all your psychometric needs! https://psychometroscar.wordpress.com/about/
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.
I don't have emotions and sometimes that makes me very sad.
for all your psychometric needs! https://psychometroscar.wordpress.com/about/
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:
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.
Tweet |