The Extended Weibull distribution

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Zhang, T., & Xie, M. (2007). Failure data analysis with extended Weibull distribution. Communications in Statistics—Simulation and Computation, 36(3), 579-592.

See also

ExW

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