ggstacy: The Generalized Gamma Distribution (Stacy's original parametrization)

Description

Probability function, distribution function, quantile function and random generation for the distribution with parameters alpha, gamma and kappa.

Usage

dggstacy(x, alpha, gamma, kappa, log = FALSE)
pggstacy(q, alpha, gamma, kappa, log.p = FALSE, lower.tail = TRUE)
qggstacy(
  p,
  alpha = 1,
  gamma = 1,
  kappa = 1,
  log.p = FALSE,
  lower.tail = TRUE,
  ...
)
rggstacy(n, alpha = 1, gamma = 1, kappa = 1, ...)

Value

dggstacy gives the (log) probability function, pggstacy gives the (log) distribution function, qggstacy gives the quantile function, and rggstacy generates random deviates.

Arguments

x: vector of (non-negative integer) quantiles.
alpha: shape parameter of the distribution (alpha > 0).
gamma: scale parameter of the distribution (gamma > 0).
kappa: shape parameter of the distribution (kappa > 0).
log, log.p: logical; if TRUE, probabilities p are given as log(p).
q: vector of quantiles.
lower.tail: logical; if TRUE (default), probabilities are $P[X \le x]$; otherwise, $P[X > x]$.
p: vector of probabilities.
...: further arguments passed to other methods.
n: number of random values to return.

Details

Probability density function: $$ f(x|\alpha, \gamma, \kappa) = \frac{\kappa}{\gamma^{\alpha}\Gamma(\alpha/\kappa)}x^{\alpha-1}\exp\left\{-\left(\frac{x}{\gamma}\right)^{\kappa}\right\}I_{[0, \infty)}(x), $$ for $\alpha>0$, $\gamma>0$ and $\kappa>0$.

Distribution function: $$ F(t|\alpha, \gamma, \kappa) = F_{G}(x|\nu, 1), $$ where $x = \displaystyle\left(\frac{t}{\gamma}\right)^\kappa$, and $F_{G}(\cdot|\nu, 1)$ corresponds to the distribution function of a gamma distribution with shape parameter $\nu = \alpha/\gamma$ and scale parameter equals to 1.