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Does anyone have a reference for mean and variance of a multivariate normal truncated along a single axis? I.e. $\mathbb{E}[X | x_i > 0]$ and $Var[X | x_i > 0]$, where $X= [x_1,..,x_n] \sim \mathcal{N}(\mu, \Sigma)$.

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Tallis explored this problem in 1961, in the publication The Moment Generating Function of the Truncated Multi-normal Distribution, published in the Journal of the Royal Statistical Society.

Later, in 1989, Leppard & Tallis published Evaluation of the mean and covariance of the truncated multinormal in Applied Statistics (38:543–553, 1989).

In 2012, Manjunath & Wilhelm published Moments Calculation For the Doubly Truncated Multivariate Normal Density, which expanded on Tallis's work which focused on single-axis truncation.

If you are in University, you likely have access to these publications via your library's subscription to JSTOR, Wiley, or other databases.

Check the R package tmvtnorm if you are interested in modeling. Link here.

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    $\begingroup$ This is just wonderfully helpful - many thanks for taking the time. $\endgroup$ – Athere Apr 5 at 0:12

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