Probability function, distribution function, quantile function and random generation for the distribution with parameters mu, sigma and varphi.
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, ...)
dggprentice gives the (log) probability function, pggprentice gives the (log) distribution function, qggprentice gives the quantile function, and rggprentice generates random deviates.
vector of (non-negative integer) quantiles.
location parameter of the distribution.
scale parameter of the distribution (sigma > 0).
shape parameter of the distribution.
logical; if TRUE, probabilities p are given as log(p).
vector of quantiles.
logical; if TRUE (default), probabilities are \(P[X \le x]\); otherwise, \(P[X > x]\).
vector of probabilities.
number of random values to return.
further arguments passed to other methods.
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 \(\sigma\).