Density, distribution function, quantile function,
random generation and hazard function for the generalized Gompertz distribution with
parameters mu sigma and nu.
Usage
dGGD(x, mu, sigma, nu, log = FALSE)
pGGD(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qGGD(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rGGD(n, mu, sigma, nu)
hGGD(x, mu, sigma, nu)Arguments
- x, q
vector of quantiles.
- mu, nu
scale parameter.
- sigma
shape parameters.
- log, log.p
logical; if TRUE, probabilities p are given as log(p).
- lower.tail
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
- p
vector of probabilities.
- n
number of observations.
Value
dGGD gives the density, pGGD gives the distribution
function, qGGD gives the quantile function, rGGD
generates random deviates and hGGD gives the hazard function.
Details
The Generalized Gompertz Distribution with parameters mu,
sigma and nu has density given by
\(f(x)= \nu \mu \exp(-\frac{\mu}{\sigma}(\exp(\sigma x - 1))) (1 - \exp(-\frac{\mu}{\sigma}(\exp(\sigma x - 1))))^{(\nu - 1)} ,\)
for \(x \geq 0\), \(\mu > 0\), \(\sigma \geq 0\) and \(\nu > 0\).
References
El-Gohary A, Alshamrani A, Al-Otaibi AN (2013). “The generalized Gompertz distribution.” Applied Mathematical Modelling, 37(1-2), 13--24.
Author
Johan David Marin Benjumea, johand.marin@udea.edu.co
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
par(mfrow = c(1, 1))
curve(dGGD(x, mu=1, sigma=0.3, nu=1.5), from = 0, to = 4,
col = "red", las = 1, ylab = "f(x)")
## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pGGD(x, mu=1, sigma=0.3, nu=1.5), from = 0, to = 4,
ylim = c(0, 1), col = "red", las = 1, ylab = "F(x)")
curve(pGGD(x, mu=1, sigma=0.3, nu=1.5, lower.tail = FALSE),
from = 0, to = 4, ylim = c(0, 1), col = "red", las = 1, ylab = "R(x)")
## The quantile function
p <- seq(from = 0, to = 0.99999, length.out = 100)
plot(x = qGGD(p=p, mu=1, sigma=0.3, nu=1.5), y = p,
xlab = "Quantile", las = 1, ylab = "Probability")
curve(pGGD(x, mu=1, sigma=0.3, nu=1.5), from = 0, add = TRUE,
col = "red")
## The random function
hist(rGGD(1000, mu=1, sigma=0.3, nu=1.5), freq = FALSE, xlab = "x",
las = 1, ylim = c(0, 0.7), main = "")
curve(dGGD(x,mu=1, sigma=0.3, nu=1.5), from = 0, to =8, add = TRUE,
col = "red")
## The Hazard function
par(mfrow=c(1,1))
curve(hGGD(x, mu=1, sigma=0.3, nu=1.5), from = 0, to = 3, col = "red",
ylab = "The hazard function", las = 1)
par(old_par) # restore previous graphical parameters