The Odd Weibull Distribution

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