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The Marshall-Olkin Extended Weibull distribution

Source: R/dMOEW.R
dMOEW.Rd

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