dist_gk: The g-and-k Distribution

In distributional: Vectorised Probability Distributions

The g-and-k Distribution

Description

The g-and-k distribution is a flexible distribution often used to model univariate data. It is particularly known for its ability to handle skewness and heavy-tailed behavior.

Usage

dist_gk(A, B, g, k, c = 0.8)

Arguments

A	Vector of A (location) parameters.
B	Vector of B (scale) parameters. Must be positive.
g	Vector of g parameters.
k	Vector of k parameters. Must be at least -0.5.
c	Vector of c parameters. 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-k random variable with parameters A, B, g, k, 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-k 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) (1 + z(u)^2)^k z(u)

where z(u) = \Phi^{-1}(u), the standard normal quantile of 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::dgk, dist_gh

Examples

dist <- dist_gk(A = 0, B = 1, g = 0, k = 0.5)
dist


mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

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

cdf(dist, 4)

quantile(dist, 0.7)

distributional documentation built on Sept. 17, 2024, 9:11 a.m.