Generalized Gamma with Location Distribution

Four-parameter Generalised Gamma distribution with location \(\mu\), scale \(s\), first shape \(\alpha\), and second shape \(\beta\). The distribution is lower-bounded at the location parameter and extends the three-parameter Stacy form by a shift. Fitting is delegated to fitlm_gengamma after subtracting the location from the data; the location is supplied by the user rather than fitted automatically. Density, distribution, quantile, and random generation functions are provided via VGAM.

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

• \(\mu\) — location parameter (location)

• \(s\) — scale parameter (scale)

• \(\alpha\) — first shape parameter (shape1)

• \(\beta\) — second shape parameter (shape2)

Usage

pgengamma_loc(q, location, scale, shape1, shape2)

dgengamma_loc(x, location, scale, shape1, shape2)

qgengamma_loc(p, location, scale, shape1, shape2)

rgengamma_loc(n, location, scale, shape1, shape2)

Arguments

location

location parameter.

scale

scale parameter.

shape1

first shape parameter.

shape2

second shape parameter.

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken as the number required.