Let
\(\mathrm{logit}(u)=
\log\left(\frac{u}{1-u}\right)\) and
\(\Lambda(z)=\frac{1}{1+e^{-z}}\),
with
\(\lambda(z)=
\Lambda(z)\left[1-\Lambda(z)\right]\).
Define
$$z =
\sqrt{\frac{1-\sigma}{\sigma}}
\left[\mathrm{logit}(x)-\mathrm{logit}(\mu)\right]
+\mathrm{logit}(\tau).$$
Cumulative distribution function
$$F(x\mid\mu,\sigma,\tau)=\Lambda(z).$$
Probability density function
$$f(x\mid\mu,\sigma,\tau)=
\sqrt{\frac{1-\sigma}{\sigma}}
\frac{\lambda(z)}{x(1-x)}.$$
Quantile function
$$Q(p\mid\mu,\sigma,\tau)=
\Lambda\!\left\{
\mathrm{logit}(\mu)
+\sqrt{\frac{\sigma}{1-\sigma}}
\left[\mathrm{logit}(p)-\mathrm{logit}(\tau)\right]
\right\}.$$
By construction \(Q(\tau)=\mu\), i.e. \(\mu\) is the \(\tau\)-th
quantile. Note that, unlike the normal-kernel Vasicek distribution, the
logistic kernel does not yield a closed-form mean; in particular
\(E(X)\neq\mu\) in general.
The GAMLSS family uses analytical derivatives. For one observation, let
$$a=\sqrt{\frac{1-\sigma}{\sigma}},\qquad
d=\mathrm{logit}(y)-\mathrm{logit}(\mu),\qquad
P=\Lambda\left\{ad+\mathrm{logit}(\tau)\right\},$$
and define
$$V=P(1-P),\qquad
b=\frac{1}{2\sigma(1-\sigma)},\qquad
g=\frac{1}{\mu(1-\mu)}.$$
If \(\ell\) denotes the individual log-likelihood contribution, the
first derivatives are
$$\frac{\partial\ell}{\partial\mu}
=-ag(1-2P)$$
and
$$\frac{\partial\ell}{\partial\sigma}
=-b\left\{1+ad(1-2P)\right\}.$$
The second and cross derivatives are
$$\frac{\partial^2\ell}{\partial\mu^2}
=g^2\left\{
a(1-2\mu)(1-2P)-2a^2V
\right\},$$
$$\frac{\partial^2\ell}{\partial\mu\,\partial\sigma}
=abg\left\{
(1-2P)-2adV
\right\},$$
and
$$\frac{\partial^2\ell}{\partial\sigma^2}
=b^2\left\{
2(1-2\sigma)
+ad(3-4\sigma)(1-2P)
-2a^2d^2V
\right\}.$$
These expressions are evaluated directly by LVASIQ(); numerical
differentiation is not used. The mean and variance
components of the family object use numerical quadrature because the
corresponding moments do not have elementary closed forms.