Generalized Gamma Distribution — GenGamma • anyFit

Three-parameter Generalised Gamma (Stacy) distribution with scale $s$, first shape $\alpha$ (shape1), and second shape $\beta$ (shape2). The distribution is supported on $x > 0$ and contains the ordinary Gamma ($\beta = 1$) and Weibull ($\beta = \alpha$) as special cases. Density, distribution, quantile, and random generation functions are provided via VGAM using the Stacy parameterisation. Fitting is performed numerically via L-BFGS-B minimisation of a normalised L-moment error function, seeded from a pre-computed lookup table (GG_InitValues) with Gauss-Legendre quadrature evaluation of the L-moments at each optimisation step.

The probability density function is: $$f(x) = \frac{\beta}{\Gamma(\alpha/\beta) \, s^\alpha} x^{\alpha-1} \exp\left(-\left(\frac{x}{s}\right)^\beta\right), \quad x > 0$$ where:

$s$ — scale parameter (scale)
$\alpha$ — first shape parameter (shape1); $k = \alpha/\beta$ is the Stacy shape
$\beta$ — second shape parameter (shape2); the distribution reduces to Gamma($\alpha$, $s$) when $\beta=1$, and to Weibull($s$, $\alpha$) when $\beta=\alpha$

Usage

dgengamma(x, scale, shape1, shape2)

pgengamma(q, scale, shape1, shape2)

qgengamma(p, scale, shape1, shape2)

rgengamma(n, scale, shape1, shape2)

Arguments

x, q: vector of quantiles.
scale: scale parameter.
shape1: first shape parameter.
shape2: second shape parameter.
p: vector of probabilities.
n: number of observations. If length(n) > 1, the length is taken as the number required.