The g-and-k Distribution

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.

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
#> <distribution[1]>
#> [1] gk(A = 0, B = 1, g = 0, k = 0.5)

mean(dist)
#> [1] 0
variance(dist)
#> [1] 4.000037
support(dist)
#> <support_region[1]>
#> [1] R
generate(dist, 10)
#> [[1]]
#>  [1]  0.53970238  2.49229280 -0.53237395  2.11935730  0.16730813  0.34230619
#>  [7] -0.65140433 -1.52376502 -1.30284695 -0.07539212
#> 

density(dist, 2)
#> [1] 0.07092933
density(dist, 2, log = TRUE)
#> [1] -2.646071

cdf(dist, 4)
#> [1] 0.9698865

quantile(dist, 0.7)
#> [1] 0.5921306