# Approximating mean/covariance of truncated/folded/censored normal distribution

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 $$\mu_X$$.