What is the difference between a truncated normal distribution and a half normal distribution in a Stochastic Frontier Analysis? I am trying to replicate a SFA where the error term u is assumed to have a cumulative normal distribution function truncated from below at zero. In my opinion, that refers to a truncated normal distribution and thus e.g. to a SFA following Battese and Coelli (1995). However, I thought of using the half-normal distribution instead, but I am not sure about the consequences. Can anyone explain what the difference between a truncated normal distribution and a half normal distribution in a Stochastic Frontier Analysis is?
 A: Partially answered in comments:
If a Normal distribution having mean = 0 is truncated from below at 0, it is (the same as) a half-Normal distribution. If the Normal distribution being truncated from below at 0 does not have mean = 0 (before the truncation), then it is not (the same as) a half-Normal distribution. – Mark L. Stone
See:  wikipedia:Truncated normal  and  wikipedia: Half-nomal 
A: "The Truncated Normal Distribution. To allow more generality into the SFM, while
guarding against distribution misspecification, a variety of one-sided distributions have been
proposed for modeling ui
in the SFM. Stevenson (1980) proposed the truncated-normal
distribution as a generalization of the half-normal distribution; whereas the half-normal
distribution is the truncation of the N(0, σ2
u
) at 0, the truncated-normal distribution is the
truncation of the N(µ, σ2
u
) at 0. The pre-truncation mean parameter, µ, affords the SFM
more flexibility in the shape of the distribution of inefficiency.
"
borrowed from "http://www.uq.edu.au/economics/cepa/docs/WP/WP022018.pdf" p. 11
