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Consider the normally distributed random vector $$X \sim \mathcal{N}(\mu, \Sigma)$$ What is the distribution of $Y = f(X)$?

For general $f$ this is a challenging problem but for the affine linear case $$f(x)_i = c_i + L_{ij}x_j$$ with $c$ a vector and $L$ a matrix. We know that this has a nice closed form. In fact, $Y$ is distributed again as a multivariate normal. $$Y \sim \mathcal{N}(c + L\mu, \;L\Sigma L^T)$$

Consider now the next simplest case. Consider that $f$ is not linear but rather quadratic. I.e. $$f(x)_i = c_i + L_{ij}x_j + H_{ijk} x_jx_k$$ with $c$ a vector, $L$ a matrix and $H$ a rank three tensor.

Does a closed form expression exist for the density of $Y$?

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  • $\begingroup$ It seems this question would have a better home on math.SE because it does not appear to be motivated by any statistical applications, but is a purely mathematical question. Is there a reason to post it here? $\endgroup$ – whuber♦ Oct 5 '11 at 14:49
  • $\begingroup$ It seemed more in the realm of statistics to me (distributions, random variables, etc....) I'm happy to move it though if you think that's best. Is there an easy method to port questions? $\endgroup$ – MRocklin Oct 5 '11 at 15:09
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    $\begingroup$ Another question about quadratic forms can be found here so maybe this question also could be considered here? $\endgroup$ – Dilip Sarwate Oct 5 '11 at 15:13
  • $\begingroup$ It's up to you, Matthew. To migrate the question, just flag it for moderator attention. It's appropriate both here and on the math site, where questions of theoretical statistics often get good answers. You'll need to decide where it might be most successful. One strategy is to leave it here for a day or two and then, if it gets no good replies, flag it for migration and try on the other site. $\endgroup$ – whuber♦ Oct 5 '11 at 15:16
  • $\begingroup$ @whuber sounds like a good plan. I'll try that out. $\endgroup$ – MRocklin Oct 5 '11 at 15:18
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The moments of such transformation can probably be found in Sec. 2.2.3 of Kollo and von Rosen (2005). Transformations of this kind have been used in some multivariate simulations. I understand there's a book on polynomials of multivariate distributions, but I have not seen it, and don't know if you'd be able to find the closed form expressions for the density of this transformation there. In a univariate case, you get a (scaled and shifted) non-central $\chi^2$ distribution, and the density expression for it is somewhat unweildy (Bessel and hypergeometric functions, or infinite series of Poisson-weighted gamma distributions).

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  • $\begingroup$ I wonder if the non-central $\chi^2$ distribution is a special case of some useful multivariate distribution. The generalizations I've seen (Wishart, etc...) don't seem appropriate. $\endgroup$ – MRocklin Oct 6 '11 at 16:26
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A related question was answered here. For that question the focus was on a particular quadratic form (the squared Euclidean norm), but this must also be one of the first things to consider.

A recommended reference is the book "Quadratic forms in random variables" by Mathai and Provost (1992, Marcel Dekker, Inc.), but I can't remember exactly how general it gets, I don't have the book, and there is no Google preview.

The results for the norm with general $\mu$ and $\Sigma$ clearly suggest that there is no simple closed form expression for the density.

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