Density, distribution function, quantile function,
random generation and hazard function for the exponentiated Weibull distribution with
parameters mu, sigma and nu.
Usage
dEW(x, mu, sigma, nu, log = FALSE)
pEW(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qEW(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rEW(n, mu, sigma, nu)
hEW(x, mu, sigma, nu)Arguments
- x, q
vector of quantiles.
- mu
scale parameter.
- sigma, nu
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
dEW gives the density, pEW gives the distribution
function, qEW gives the quantile function, rEW
generates random deviates and hEW gives the hazard function.
Details
The Exponentiated Weibull Distribution with parameters mu,
sigma and nu has density given by
\(f(x)=\nu \mu \sigma x^{\sigma-1} \exp(-\mu x^\sigma) (1-\exp(-\mu x^\sigma))^{\nu-1},\)
for \(x > 0\), \(\mu > 0\), \(\sigma > 0\) and \(\nu > 0\).
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dEW(x, mu=2, sigma=1.5, nu=0.5), from=0, to=2,
ylim=c(0, 2.5), col="red", las=1, ylab="f(x)")
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pEW(x, mu=2, sigma=1.5, nu=0.5),
from=0, to=2, col="red", las=1, ylab="F(x)")
curve(pEW(x, mu=2, sigma=1.5, nu=0.5, lower.tail=FALSE),
from=0, to=2, col="red", las=1, ylab="R(x)")
## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qEW(p, mu=2, sigma=1.5, nu=0.5), y=p, xlab="Quantile",
las=1, ylab="Probability")
curve(pEW(x, mu=2, sigma=1.5, nu=0.5), from=0, add=TRUE, col="red")
## The random function
hist(rEW(n=10000, mu=2, sigma=1.5, nu=0.5), freq=FALSE,
xlab="x", las=1, main="")
curve(dEW(x, mu=2, sigma=1.5, nu=0.5), from=0, add=TRUE, col="red")
## The Hazard function
par(mfrow=c(1,1))
curve(hEW(x, mu=2, sigma=1.5, nu=0.5), from=0, to=2, ylim=c(0, 7),
col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters