3-parameter Gamma Distribution

Three-parameter Gamma (Pearson type III) distribution with location \(\mu\), scale \(\beta\), and shape \(\alpha\). The distribution is lower-bounded at the location parameter and is widely employed in flood-frequency analysis. Density, distribution, quantile, and random generation functions are provided via the PearsonDS package. Fitting is performed via closed-form L-moments through pelpe3.

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

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

• \(\beta\) — scale parameter (scale)

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

Usage

dgamma3(x, location, scale, shape)

pgamma3(q, location, scale, shape)

qgamma3(p, location, scale, shape)

rgamma3(n, location, scale, shape)

Arguments

x, q

vector of quantiles.

location

location parameter.

scale

scale parameter.

shape

shape parameter.

p

vector of probabilities.

n

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