The Generalized Gamma Distribution (Prentice's alternative parametrization)

Probability function, distribution function, quantile function and random generation for the distribution with parameters mu, sigma and varphi.

Usage

dggprentice(x, mu, sigma, varphi, log = FALSE)

pggprentice(q, mu = 0, sigma = 1, varphi, lower.tail = TRUE, log.p = FALSE)

qggprentice(p, mu = 0, sigma = 1, varphi, lower.tail = TRUE, log.p = FALSE)

rggprentice(n, mu = 0, sigma = 1, varphi, ...)

Arguments

x

vector of (non-negative integer) quantiles.

mu

location parameter of the distribution.

sigma

scale parameter of the distribution (sigma > 0).

varphi

shape parameter of the distribution.

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.

n

number of random values to return.

...

further arguments passed to other methods.

Value

dggprentice gives the (log) probability function, pggprentice gives the (log) distribution function, qggprentice gives the quantile function, and rggprentice generates random deviates.

Details

Probability density function: $$ f(x | \mu, \sigma, \varphi) = \begin{cases} \frac{|\varphi|(\varphi^{-2})^{\varphi^{-2}}}{\sigma x\Gamma(\varphi^{-2})}\exp\{\varphi^{-2}[\varphi w - \exp(\varphi w)]\}I_{[0, \infty)}(x), & \varphi \neq 0 \\ \frac{1}{\sqrt{2\pi}x\sigma}\exp\left\{-\frac{1}{2}\left(\frac{log(x)-\mu}{\sigma}\right)^2\right\}I_{[0, \infty)}(x), & \varphi = 0 \end{cases} $$ where \(w = \frac{\log(x) - \mu}{\sigma}\), for \(-\infty < \mu < \infty\), \(\sigma>0\) and \(-\infty < \varphi < \infty\).

Distribution function: $$ F(x|\mu, \sigma, \varphi) = \begin{cases} F_{G}(y|1/\varphi^2, 1), & \varphi > 0 \\ 1-F_{G}(y|1/\varphi^2, 1), & \varphi < 0 \\ F_{LN}(x|\mu, \sigma), & \varphi = 0 \end{cases} $$ where \(y = \displaystyle\left(\frac{x}{\sigma}\right)^\varphi\), \(F_{G}(\cdot|\nu, 1)\) is the distribution function of a gamma distribution with shape parameter \(1/\varphi^2\) and scale parameter equals to 1, and \(F_{LN}(x|\mu, \sigma)\) corresponds to the distribution function of a lognormal distribution with location parameter \(\mu\) and scale parameter \(\sigma\).