The probability is computed as an expectation over the inefficiency term,
$$F(q) = E_u\left[F_v\left((q + su)/\sigma_v\right)\right],$$
with \(s = +1\) for production and \(-1\) for cost. Conditioning on
\(u\) leaves the noise CDF in closed form, so the noise is never integrated
and the tails inherit pnorm()'s (or pt()'s) own accuracy.
Three parameterizations do not read the way their names suggest, and are
taken from each likelihood in sfm() rather than from the label:
"THT" lists sigu before sigv, the only model that
does; "NG"'s sigu is the gamma scale and its mu
the shape; "NNAK"'s sigu is the Nakagami spread, so
\(\Omega = \code{sigu}^2\); and "NLN"'s mu is a
meanlog.
"THT" is not an independent convolution. Tancredi's (2002)
composed error is skew-\(t\), a scale mixture in which \(v\) and \(u\)
are divided by the same \(\sqrt{V/a}\). Treating it as
\(t\) noise plus an independent half-normal -- which is what "tHN"
actually is -- gets the log density wrong by up to 1.74, so the two models
take different paths here despite looking alike.
Verified two ways: against pcomposed for the half-normal case,
agreeing to \(3.5\times10^{-15}\) relative including a lower tail at
\(\log F = -209\); and for all thirteen models against a four-million-draw
simulation, with a maximum absolute error of \(4\times10^{-4}\) against a
Monte Carlo standard error of \(7.5\times10^{-4}\). Numerically
differentiating this CDF reproduces each model's own stored log-density to
about \(10^{-8}\), except for the simulated-ML models "NLN" and
"NW", where the difference is the simulation error of their
likelihood rather than a disagreement about the model.