# Is the variance of the multivariate folded normal distribution known?

The mean and variance of the folded normal distribution are known. Consider now the distribution of $(|x_1|, \ldots, |x_n|)$, where $\mathbb{x} \sim N(\mu, \Sigma)$. The mean of the multivariate folded normal distribution is easy to obtain. But what about the variance? I could not find references, and the calculation seems complicated. Any suggestion?

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If $x = (x_1, x_2, \ldots, x_n)$, does $|x|$ equal $(|x_1|, |x_2|, \ldots, |x_n|)$ or $\sqrt{x_1^2+x_2^2+\cdots+x_n^2}$? If the former, are you asking for the variance of each $|x_i|$ or the covariance matrix of the random vector? –  Dilip Sarwate Jan 28 '12 at 4:32
@DilipSarwate thanks for the question. Clarification added. –  gappy Jan 28 '12 at 14:00
Do you want just the covariance matrix or the actual multivariate distribution? –  cardinal Jan 28 '12 at 16:54

There is a section entitled 'Bivariate Half-normal distribution in: Continuous Multivariate Distributions: Models and applications By Samuel Kotz, Norman Lloyd Johnson, N. Balakrishnan.

I would be curious to see how this can be generalized to a random vector of any dimensions.

In fact, the bivariate case appears to be thoroughly treated in this paper: http://www.stat-athens.aueb.gr/~jpan/papers/Panaretos-ApplStatScience2001(119-136)ft.pdf

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I don't know what you mean by folded normal distribution. The distribution of $|X|$ where $X \sim N(0,1)$? The distribution of $|X|$ when $X \sim N(\mu,\sigma^2)$? But, regardless of the interpretation, if you aver that "The mean and variance of the folded normal distribution are known" to you, then rest assured that if $x \sim N(\mu,\Sigma)$ has a multivariate normal distribution, then $x_i \sim N(\mu_i, \Sigma_{i,i})$ and so whatever formulas are known to you as the mean and variance of $|X|$ where $X \sim N(\mu,\sigma^2)$ also can be used for the mean and variance of $|x_i|$ which has a folded normal distribution since $x_i \sim N(\mu_i, \Sigma_{i,i})$.
• If you know only the mean and variance of $|X|$ when $X \sim N(0,1)$ but not when $X \sim N(\mu,\sigma^2)$, then please edit your question to say so clearly.
• If you know formulas for the mean and variance of $|X|$ where $X \sim N(\mu,\sigma^2)$, please apply the formulas to each $|x_i|$ since $x_i \sim N(\mu_i, \sigma_{i,i})$. It would probably help the readers of this forum of you were to type in the formulas for the mean and variance of $|X|$.
• If you want to know the covariance of $|x_i|$ and $|x_j|$, please edit your question to say so clearly. You have been asked the same question by cardinal also.
The formulas for the univariate folded normal are known. The same formulas can be applied for the variance of $|x_i|$, since $x_i$ is normal. I am stuck at computing $E|x_i x_j|$. –  gappy Feb 1 '12 at 2:35