Density, distribution function, quantile function,
random generation and hazard function for the Marshall-Olkin Extended Weibull distribution
with parameters mu, sigma and nu.
Usage
dMOEW(x, mu, sigma, nu, log = FALSE)
pMOEW(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qMOEW(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rMOEW(n, mu, sigma, nu)
hMOEW(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
dMOEW gives the density, pMOEW gives the distribution
function, qMOEW gives the quantile function, rMOEW
generates random deviates and hMOEW gives the hazard function.
Details
The Marshall-Olkin Extended Weibull distribution mu,
sigma and nu has density given by
\(f(x) = \frac{\mu \sigma \nu (\nu x)^{\sigma - 1} exp\{{-(\nu x )^{\sigma}}\}}{\{1-(1-\mu) exp\{{-(\nu x )^{\sigma}}\} \}^{2}},\)
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 .
M.E G, E.K A, R.A J (2005). “Marshall–Olkin extended weibull distribution and its application to censored data.” Journal of Applied Statistics, 32(10), 1025--1034. doi:10.1080/02664760500165008 .
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(dMOEW(x, mu=0.5, sigma=0.7, nu=1), from=0.001, to=1,
col="red", ylab="f(x)", las=1)
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pMOEW(x, mu=0.5, sigma=0.7, nu=1),
from=0.0001, to=2, col="red", las=1, ylab="F(x)")
curve(pMOEW(x, mu=0.5, sigma=0.7, nu=1, 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=qMOEW(p, mu=0.5, sigma=0.7, nu=1), y=p, xlab="Quantile",
las=1, ylab="Probability")
curve(pMOEW(x, mu=0.5, sigma=0.7, nu=1),
from=0, add=TRUE, col="red")
## The random function
hist(rMOEW(n=100, mu=0.5, sigma=0.7, nu=1), freq=FALSE,
xlab="x", ylim=c(0, 1), las=1, main="")
curve(dMOEW(x, mu=0.5, sigma=0.7, nu=1),
from=0.001, to=2, add=TRUE, col="red")
## The Hazard function
curve(hMOEW(x, mu=0.5, sigma=0.7, nu=1), from=0.001, to=3,
col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters
