These are the pieces a declared non-normal random effect is built
from (see the dist() line of an ini({}) block). The latent random
effect stays standard normal, phiU() maps it to a uniform, and the
family's inverse CDF maps that uniform to the random effect.
Usage
phiU(q)
ibeta(a, b, x)
ibetaDer(a, b, x)
ibetaInv(a, b, p)
ibetaDera(a, b, x)
ibetaDerb(a, b, x)
gammapDera(a, x)
studentTDen(x, nu)
studentTCdf(x, nu)
studentTCdfDnu(x, nu)
studentTInv(p, nu)Details
phiU() is phi() bounded away from 0 and 1 by 1e-15. phi()
saturates to exactly 0 or 1 in double precision around |q| = 8.3,
where an inverse CDF would return an infinity.
ibeta(), ibetaDer() and ibetaInv() are the regularized
incomplete beta function, its derivative in x and its inverse – ie
pbeta(), dbeta() and qbeta(). studentTCdf(), studentTDen()
and studentTInv() are the Student t CDF, density and quantile,
written on the incomplete beta so the quantile is exactly the
inverse of the CDF at the same tolerance.
gammapDera(), ibetaDera(), ibetaDerb() and studentTCdfDnu()
are the derivatives of those CDFs with respect to their SHAPE
parameters. None has an elementary closed form, so they are central
differences with one Richardson extrapolation. They exist so that
rxode2's derivative table is complete for these functions: without
them a model using an inverse CDF silently degrades to a one sided
finite difference, which is precisely wrong in the tails.
Examples
phiU(c(-9, 0, 9))
#> [1] 1e-15 5e-01 1e+00
ibetaInv(2, 3, 0.5) ## == qbeta(0.5, 2, 3)
#> [1] 0.3857276
studentTInv(0.975, 6) ## == qt(0.975, 6)
#> [1] 2.446912