Density, distribution function, quantile function,
random generation and hazard function for the generalized
modified weibull distribution with parameters mu,
sigma, nu and tau.
Usage
dGMW(x, mu, sigma, nu, tau, log = FALSE)
pGMW(q, mu, sigma, nu, tau, lower.tail = TRUE, log.p = FALSE)
qGMW(p, mu, sigma, nu, tau, lower.tail = TRUE, log.p = FALSE)
rGMW(n, mu, sigma, nu, tau)
hGMW(x, mu, sigma, nu, tau, log = FALSE)Arguments
- x, q
vector of quantiles.
- mu
scale parameter.
- sigma
shape parameter.
- nu
shape parameter.
- tau
acceleration 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
dGMW gives the density, pGMW gives the distribution
function, qGMW gives the quantile function, rGMW
generates random deviates and hGMW gives the hazard function.
Details
The generalized modified weibull with parameters mu,
sigma, nu and tau has density given by
\(f(x)= \mu \sigma x^{\nu - 1}(\nu + \tau x) \exp(\tau x - \mu x^{\nu} e^{\tau x}) [1 - \exp(- \mu x^{\nu} e^{\tau x})]^{\sigma-1},\)
for x>0.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5), from=0, to=0.8,
ylim=c(0, 2), col="red", las=1, ylab="f(x)")
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5),
from=0, to=1.2, col="red", las=1, ylab="F(x)")
curve(pGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5, lower.tail=FALSE),
from=0, to=1.2, col="red", las=1, ylab="R(x)")
## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qGMW(p, mu=2, sigma=0.5, nu=2, tau=1.5), y=p, xlab="Quantile",
las=1, ylab="Probability")
curve(pGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5), from=0, add=TRUE, col="red")
## The random function
hist(rGMW(n=1000, mu=2, sigma=0.5, nu=2,tau=1.5), freq=FALSE,
xlab="x", main ="", las=1)
curve(dGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5), from=0, add=TRUE, col="red")
## The Hazard function
par(mfrow=c(1,1))
curve(hGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5), from=0, to=1, ylim=c(0, 16),
col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters