Density, distribution function, quantile function,
random generation and hazard function for the Extended Weibull distribution
with parameters mu, sigma and nu.
Usage
dExW(x, mu, sigma, nu, log = FALSE)
pExW(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qExW(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rExW(n, mu, sigma, nu)
hExW(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
dExW gives the density, pExW gives the distribution
function, qExW gives the quantile function, rExW
generates random deviates and hExW gives the hazard function.
Details
The Extended Weibull distribution with parameters mu,
sigma and nu has density given by
\(f(x) = \frac{\mu \sigma \nu x^{\sigma -1} exp({-\mu x^{\sigma}})} {[1 -(1-\nu) exp({-\mu x^{\sigma}})]^2},\)
for x > 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 .
Tieling Z, Min X (2007). “Failure Data Analysis with Extended Weibull Distribution.” Communications in Statistics - Simulation and Computation, 36, 579--592. doi:10.1080/03610910701236081 .
Author
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dExW(x, mu=0.3, sigma=2, nu=0.05), from=0.0001, to=2,
col="red", las=1, ylab="f(x)")
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pExW(x, mu=0.3, sigma=2, nu=0.05),
from=0.0001, to=2, col="red", las=1, ylab="F(x)")
curve(pExW(x, mu=0.3, sigma=2, nu=0.05, lower.tail=FALSE),
from=0.0001, to=2, col="red", las=1, ylab="R(x)")
## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qExW(p, mu=0.3, sigma=2, nu=0.05), y=p, xlab="Quantile",
las=1, ylab="Probability")
curve(pExW(x, mu=0.3, sigma=2, nu=0.05),
from=0, add=TRUE, col="red")
## The random function
hist(rExW(n=10000, mu=0.3, sigma=2, nu=0.05), freq=FALSE,
xlab="x", ylim=c(0, 2), las=1, main="")
curve(dExW(x, mu=0.3, sigma=2, nu=0.05),
from=0.001, to=4, add=TRUE, col="red")
## The Hazard function
curve(hExW(x, mu=0.3, sigma=2, nu=0.05), from=0.001, to=4,
col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters
