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I am studying Elliptically Symmetric Distributions and someone recommended me Symmetric Multivariate and Related Distributions by Fang (1990), which is the book I am reading for that purpose.

I know that if $X\sim N(\mu,\sigma^2)$, then $|X|$ follows a folded normal distribution. So, I wonder if there exists some generalization. For that, let $X\sim \operatorname{EC}_1(\mu,\sigma^2,\psi)$, what is the distribution of $|X|$?

I thought using PDF's definition of $X\sim \operatorname{EC}_1(\mu,\sigma^2,\psi)$: $$f(x)=\frac{1}{\sigma}\psi\left(\frac{(x-\mu)^2}{\sigma^2}\right)$$ But I think it is the wrong way. Do you know what is the distribution of $|X|$ to know what I should have to prove? Or if you know an article that talks about this, It would be really helpful.

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To my knowledge, this is still under active research, I am not aware of a generalization to elliptical distributions.

The latest article that I know of is On Moments of Folded and Truncated Multivariate Normal Distributions, which states in the conclusion:

Generalizing the results to multivariate elliptical distributions requires a lot more work. Although the product moments of multivariate elliptical distributions can be obtained from the product moments of multivariate normal distributions (see, for example, Berkane and Bentler (1986) and Maruyama and Seo (2003)), it is not clear how to obtain product moments of folded and truncated multivariate elliptical distributions. We leave this topic for future research.

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