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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$\begingroup$ 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? $\endgroup$ – Dilip Sarwate Jan 28 '12 at 4:32
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$\begingroup$ @DilipSarwate thanks for the question. Clarification added. $\endgroup$ – gappy Jan 28 '12 at 14:00
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$\begingroup$ Do you want just the covariance matrix or the actual multivariate distribution? $\endgroup$ – cardinal Jan 28 '12 at 16:54
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$\begingroup$ Since this question was asked, a paper (sankhya.isical.ac.in/search/75b1/13571_2013_64_PrintPDF.pdf ) on the multivariate folded normal came out. It is not clear if this matches your setting. $\endgroup$ – shabbychef Sep 8 '14 at 17:34
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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.
answered Jan 28 '12 at 17:40
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$\begingroup$ 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 $E|x_i x_j|$. $\endgroup$ – gappy Feb 1 '12 at 2:35
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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
