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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https://stats.stackexchange.com/questions/163172 and https://stats.stackexchange.com/questions/385423 answer the question about the mean. The variance question was asked but not answered at https://stats.stackexchange.com/questions/330771. https://stats.stackexchange.com/questions/504245/ (closed) has a list of links to answers to closely related questions. The full distribution (not just mean and variance) is obtained for an equivalent question (with slightly different formulation) at https://stats.stackexchange.com/questions/444925. – whuber Apr 02 '21 at 16:54
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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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