Density, distribution function, quantile function,
random generation and hazard function for the modified weibull distribution
with parameters mu, sigma and nu.
Usage
dMW(x, mu, sigma, nu, log = FALSE)
pMW(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qMW(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rMW(n, mu, sigma, nu)
hMW(x, mu, sigma, nu)Arguments
- x, q
vector of quantiles.
- mu
shape parameter one.
- sigma
parameter two.
- nu
scale parameter three.
- 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
dMW gives the density, pMW gives the distribution
function, qMW gives the quantile function, rMW
generates random deviates and hMW gives the hazard function.
Details
The modified weibull distribution with parameters mu, sigma
and nu has density given by
\(f(x) = \mu (\sigma + \nu x) x^{\sigma - 1} \exp(\nu x) \exp(-\mu x^{\sigma} \exp(\nu x))\)
for \(x > 0\), \(\mu > 0\), \(\sigma \geq 0\) and \(\nu \geq 0\).
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 .
Lai CD, Xie M, Murthy DNP (2003). “A modified Weibull distribution.” IEEE Transactions on reliability, 52(1), 33--37.
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(dMW(x, mu=2, sigma=1.5, nu=0.2), from=0, to=2,
ylim=c(0, 1.5), col="red", las=1, ylab="f(x)")
## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pMW(x, mu=2, sigma=1.5, nu=0.2), from=0, to=2,
col = "red", las=1, ylab="F(x)")
curve(pMW(x, mu=2, sigma=1.5, nu=0.2, lower.tail = FALSE),
from=0, to=2, col="red", las=1, ylab ="R(x)")
## The quantile function
p <- seq(from=0, to=0.9999, length.out=100)
plot(x=qMW(p, mu=2, sigma=1.5, nu=0.2), y=p, xlab="Quantile",
las=1, ylab="Probability")
curve(pMW(x, mu=2, sigma=1.5, nu=0.2), from=0, add=TRUE, col="red")
## The random function
hist(rMW(n=1000, mu=2, sigma=1.5, nu=0.2), freq=FALSE,
xlab="x", las=1, main="")
curve(dMW(x, mu=2, sigma=1.5, nu=0.2), from=0, add=TRUE, col="red")
## The Hazard function
par(mfrow=c(1,1))
curve(hMW(x, mu=2, sigma=1.5, nu=0.2), from=0, to=1.5, ylim=c(0, 5),
col="red", las=1, ylab="H(x)", las=1)
par(old_par) # restore previous graphical parameters