Density, distribution function, quantile function,
random generation and hazard function for the Gamma Weibull distribution
with parameters mu, sigma, nu and tau.
Usage
dGammaW(x, mu, sigma, nu, log = FALSE)
pGammaW(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qGammaW(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rGammaW(n, mu, sigma, nu)
hGammaW(x, mu, sigma, nu)Arguments
- x, q
vector of quantiles.
- mu
parameter.
- sigma
parameter.
- nu
parameter.
- 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
dGammaW gives the density, pGammaW gives the distribution
function, qGammaW gives the quantile function, rGammaW
generates random deviates and hGammaW gives the hazard function.
Details
The Gamma Weibull Distribution with parameters mu,
sigma and nu has density given by
\(f(x)= \frac{\sigma \mu^{\nu}}{\Gamma(\nu)} x^{\nu \sigma - 1} \exp(-\mu x^\sigma),\)
for \(x > 0\), \(\mu > 0\), \(\sigma \geq 0\) and \(\nu > 0\).
References
Almalki SJ, Nadarajah S (2014). “Modifications of the Weibull distribution: A review.” Reliability Engineering & System Safety, 124, 32--55. doi:10.1016/j.ress.2013.11.010 .
Stacy EW, others (1962). “A generalization of the gamma distribution.” The Annals of mathematical statistics, 33(3), 1187--1192.
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
curve(dGammaW(x, mu = 0.5, sigma = 2, nu=1), from = 0, to = 6,
col = "red", las = 1, ylab = "f(x)")
## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pGammaW(x, mu = 0.5, sigma = 2, nu=1), from = 0, to = 3,
ylim = c(0, 1), col = "red", las = 1, ylab = "F(x)")
curve(pGammaW(x, mu = 0.5, sigma = 2, nu=1, lower.tail = FALSE),
from = 0, to = 3, 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 = qGammaW(p = p, mu = 0.5, sigma = 2, nu=1), y = p,
xlab = "Quantile", las = 1, ylab = "Probability")
curve(pGammaW(x, mu = 0.5, sigma = 2, nu=1), from = 0, add = TRUE,
col = "red")
## The random function
hist(rGammaW(1000, mu = 0.5, sigma = 2, nu=1), freq = FALSE, xlab = "x",
ylim = c(0, 1), las = 1, main = "")
curve(dGammaW(x, mu = 0.5, sigma = 2, nu=1), from = 0, add = TRUE,
col = "red", ylim = c(0, 1))
## The Hazard function
par(mfrow=c(1,1))
curve(hGammaW(x, mu = 0.5, sigma = 2, nu=1), from = 0, to = 2,
ylim = c(0, 1), col = "red", ylab = "Hazard function", las = 1)
par(old_par) # restore previous graphical parameters