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Given a normally distributed $X$, what is the best way to approximate the covariance matrix and mean vector of $\tilde{X} = \max(0, X)$? I am interested in the censored distribution, but the truncated or folded case should not be that much different.

The problem is that the covariance matrix is quite big $n \approx 500$. For small $n$ the recurrence relation described in On Moments of Folded and Truncated Multivariate Normal Distributions works quite well, but not for $n \gg 5$.

I am looking for an alternative to just using the diagonal matrix or a Taylor approximation.

The first-order Taylor approximation for the mean is $E[\max(0, X)] \approx \max(0, \mu_X)$. However, I am wondering whether this approximation is that good. According to Wikipedia the mean should be greater than $\mu_X$, but the approximation just sets a cutting point at $0$.

For $\Sigma_X = \begin{bmatrix}1 & 0.1\\0.1 & 1\end{bmatrix}$ and $\boldsymbol{\mu_X} = 0$, the exact censored mean is $\boldsymbol{\mu_\tilde{X}} \approx 0.219418$. A diagonal approximation (ignoring correlations) gives me $\boldsymbol{\mu_\tilde{X}} \approx 0.398942$. So this is as expected higher than $\mu_X$.

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