ggprentice | R Documentation |
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, ...)
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 |
vector of probabilities. |
n |
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
.
dggprentice gives the (log) probability function, pggprentice gives the (log) distribution function, qggprentice gives the quantile function, and rggprentice generates random deviates.
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