Density, distribution function, quantile function,
random generation and hazard function for the Odd Weibull distribution with
parameters mu, sigma and nu.
Usage
dOW(x, mu, sigma, nu, log = FALSE)
pOW(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qOW(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rOW(n, mu, sigma, nu)
hOW(x, mu, sigma, nu)Arguments
- x, q
vector of quantiles.
- mu
parameter one.
- sigma
parameter two.
- nu
parameter three.
- log, log.p
logical; if TRUE, probabilities p are given as log(p).
- lower.tail
logical; if TRUE (default), probabilities are P[T <= t], otherwise, P[T > t].
- p
vector of probabilities.
- n
number of observations.
Value
dOW gives the density, pOW gives the distribution
function, qOW gives the quantile function, rOW
generates random deviates and hOW gives the hazard function.
Details
The Odd Weibull with parameters mu, sigma
and nu has density given by
\(f(x) = \left( \frac{\sigma\nu}{x} \right) (\mu x)^\sigma e^{(\mu x)^\sigma} \left(e^{(\mu x)^{\sigma}}-1\right)^{\nu-1} \left[ 1 + \left(e^{(\mu x)^{\sigma}}-1\right)^\nu \right]^{-2}\)
for x > 0.
References
Cooray K (2006). “Generalization of the Weibull distribution: The odd Weibull family.” Statistical Modelling, 6(3), 265--277. ISSN 1471082X, doi:10.1191/1471082X06st116oa .
Author
Jaime Mosquera Gutiérrez jmosquerag@unal.edu.co
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dOW(x, mu=2, sigma=3, nu=0.2), from=0, to=4, ylim=c(0, 2),
col="red", las=1, ylab="f(x)")
## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pOW(x, mu=2, sigma=3, nu=0.2), from=0, to=4, ylim=c(0, 1),
col="red", las=1, ylab="f(x)")
curve(pOW(x, mu=2, sigma=3, nu=0.2, lower.tail=FALSE), from=0,
to=4, ylim=c(0, 1), col="red", las=1,
ylab = "R(x)")
## The quantile function
p <- seq(from=0, to=0.998, length.out=100)
plot(x = qOW(p, mu=2, sigma=3, nu=0.2), y=p, xlab="Quantile", las=1,
ylab="Probability")
curve(pOW(x, mu=2, sigma=3, nu=0.2), from=0, add=TRUE, col="red")
## The random function
hist(rOW(n=10000, mu=2, sigma = 3, nu = 0.2), freq=FALSE, ylim = c(0, 2),
xlab = "x", las = 1, main = "")
curve(dOW(x, mu=2, sigma=3, nu=0.2), from=0, ylim=c(0, 2), add=TRUE,
col = "red")
## The Hazard function
par(mfrow=c(1,1))
curve(hOW(x, mu=2, sigma=3, nu=0.2), from=0, to=2.5, ylim=c(0, 3),
col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters