Is the variance of the multivariate folded normal distribution known?

Question

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?

Answer 1

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

Answer 2

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.