Let $\mathbf{X} \sim \mathcal{N}_n( \mathbf{m}, \mathbf{C})$ be an $n$-dimensional gaussian vector, where $\mathbf{C} \in \mathbb{R}^{n \times n}$ is not diagonal, but it is positive-definitive, $\mathbf{C} \succ 0$, and $\mathbf{m} \neq \mathbf{0}$. Let $$ Y = \| \mathbf{X} \|^2 $$ where $\| \cdot \|$ denotes the $L_2$-norm (Euclidean norm), should follow a generalized chi-squared distribution. Unfortunately, the Wikipedia page contains little information.

My question is: What is the PDF (probability density function) of $Y$ ?


1 Answer 1


Your question is really a special case of https://math.stackexchange.com/questions/442472/sum-of-squares-of-dependent-gaussian-random-variables/442916#442916 (with $A=I$).

But nevertheless: $X$ is multivariate normal ($n$ components) with expectation $m$ and positive definite covariance matrix $C$. We are interested in the distribution of $\| X\|^2 = X^T X$.

Define $W = C^{-1/2}(X-m)$. Then $W$ is multivariate normal with expectation zero and covariance matrix $I$, the identity matrix. $X= C^{1/2}W +m$, so after some algebra $$ X^T X= (W+C^{-1/2}m)^T C (W+C^{-1/2}m) $$ Use the spectral theorem to write $C=P^T\Lambda P$ where $P$ is an orthogonal matrix (such that $P^T P = P P^T = I$) and $\Lambda$ is a diagonal matrix with positive diagonal elements $\lambda_1, \dotsc, \lambda_n$. Write $U=PW$, $U$ is also multivariate normal with mean zero and identity covariance matrix. Now, with some algebra we find that $$ X^TX= (U+b)^T \Lambda (U+b) = \sum_{j=1}^n \lambda_j (U_j+b_j)^2 $$ where $b_j= \Lambda^{-1/2} P m$, so that $X^TX$ is a linear combination of independent noncentral chisquare variables, each with one degree of freedom and noncentrality $b_j^2$. Except for special cases, it would be hard to find a closed exact expression for its density function (for instance, if all $\lambda_j$ are equal, it will be a constant times a noncentral chisquare). For some ideas which could be used, in particular saddlepoint approximation, see the posts Generic sum of Gamma random variables, How does saddlepoint approximation work? and for the needed moment generating functions, What is the moment generating function of the generalized (multivariate) chi-square distribution?

There is a book-length treatment by Mathai and Provost https://books.google.no/books/about/Quadratic_Forms_in_Random_Variables.html?id=tFOqQgAACAAJ&redir_esc=y about quadratic forma in random variables. It gives a lot of different approximations, typically series expansions. There are also some exact (very complicated) results, but only for some special cases. I would go for the saddlepoint approximation! (I will try to come back and post some examples here, but not tonight ...)

There is also an R package https://CRAN.R-project.org/package=CompQuadForm with some approximations.

  • $\begingroup$ If I am reading your other answer correctly, the MGF of $\| \mathbf{X} \|^2$ should have a closed-form, a pretty simple one actually. You should edit your answer and post the MGF here. The PDF, then, is easily derived from the MGF. $\endgroup$ Apr 12, 2018 at 10:49
  • $\begingroup$ Regarding approximations, unfortunately, I need exact results. Approximations were useful in 1992 (year of the book by Mathai and Provost) when computers were old and slow. Now, when computers can calculate hypergeometric functions in microseconds (or even less), they have lost their purpose. $\endgroup$ Apr 12, 2018 at 10:51
  • $\begingroup$ Why do you need exact results? I will post the mgf here and see into inversion, but when I have some time ... $\endgroup$ Apr 12, 2018 at 12:51
  • $\begingroup$ I need exact results because I need to see what happens in more complicated problems than the simple one presented here. Also, I need to manipulate the PDF itself and, most likely, the CDF will also be involved. $\endgroup$ Apr 13, 2018 at 15:39
  • $\begingroup$ But do you need it in the most general case, or could some special cases do? (like sum of only two variables, ...) $\endgroup$ Apr 13, 2018 at 16:06

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