We recommend reading this documentation on
https://pkg.mitchelloharawild.com/distributional/, where the math
will render nicely.
In the following, let \(X\) be a g-and-h random variable with parameters
A
, B
, g
, h
, and c
.
Support: \((-\infty, \infty)\)
Mean: Not available in closed form.
Variance: Not available in closed form.
Probability density function (p.d.f):
The g-and-h distribution does not have a closed-form expression for its density. Instead,
it is defined through its quantile function:
$$
Q(u) = A + B \left( 1 + c \frac{1 - \exp(-gz(u))}{1 + \exp(-gz(u))} \right) \exp(h z(u)^2/2) z(u)
$$
where \(z(u) = \Phi^{-1}(u)\)
Cumulative distribution function (c.d.f):
The cumulative distribution function is typically evaluated numerically due to the lack
of a closed-form expression.