ggprentice: The Generalized Gamma Distribution (Prentice's alternative...

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The Generalized Gamma Distribution (Prentice's alternative parametrization)

Description

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.

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.

Value

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

survstan documentation built on May 29, 2024, 8:41 a.m.