distributional (version 0.5.0)

dist_gh: The generalised g-and-h Distribution

Description

The generalised g-and-h distribution is a flexible distribution used to model univariate data, similar to the g-k distribution. It is known for its ability to handle skewness and heavy-tailed behavior.

Usage

dist_gh(A, B, g, h, c = 0.8)

Arguments

A

Vector of A (location) parameters.

B

Vector of B (scale) parameters. Must be positive.

g

Vector of g parameters.

h

Vector of h parameters. Must be non-negative.

c

Vector of c parameters (used for generalised g-and-h). Often fixed at 0.8 which is the default.

Details

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: $(-\mathrm{\infty },\mathrm{\infty })$

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:

$$Q(u)=A+B(1+c\frac{1-\exp (-gz(u))}{1+\exp (-gz(u))})\exp (hz(u{)}^2/2)z(u)$$

where $z(u)={\mathrm{\Phi }}^{-1}(u)$

Cumulative distribution function (c.d.f):

The cumulative distribution function is typically evaluated numerically due to the lack of a closed-form expression.

See Also

gk::dgh, dist_gk

Examples

dist <- dist_gh(A = 0, B = 1, g = 0, h = 0.5)
dist

if (FALSE) { # requireNamespace("gk", quietly = TRUE)
mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)
}