dEW function

The Exponentiated Weibull distribution

Density, distribution function, quantile function, random generation and hazard function for the exponentiated Weibull distribution with parameters mu, sigma and nu.

dEW(x, mu, sigma, nu, log = FALSE)

pEW(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)

qEW(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)

rEW(n, mu, sigma, nu)

hEW(x, mu, sigma, nu)

Arguments

• x, q: vector of quantiles.
• mu: scale parameter.
• sigma, nu: shape parameters.
• 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.

Returns

dEW gives the density, pEW gives the distribution function, qEW gives the quantile function, rEW

generates random deviates and hEW gives the hazard function.

Details

The Exponentiated Weibull Distribution with parameters mu, sigma and nu has density given by

$f(x)=\nu \mu \sigma x^{\sigma-1} \exp(-\mu x^\sigma) (1-\exp(-\mu x^\sigma))^{\nu-1},$

for $x > 0$, $\mu > 0$, $\sigma > 0$ and $\nu > 0$.

Examples

old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters

## The probability density function
curve(dEW(x, mu=2, sigma=1.5, nu=0.5), from=0, to=2,
      ylim=c(0, 2.5), col="red", las=1, ylab="f(x)") 

## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pEW(x, mu=2, sigma=1.5, nu=0.5), 
      from=0, to=2,  col="red", las=1, ylab="F(x)")
curve(pEW(x, mu=2, sigma=1.5, nu=0.5, lower.tail=FALSE), 
      from=0, to=2, col="red", las=1, ylab="R(x)")

## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qEW(p, mu=2, sigma=1.5, nu=0.5), y=p, xlab="Quantile",
     las=1, ylab="Probability")
curve(pEW(x, mu=2, sigma=1.5, nu=0.5), from=0, add=TRUE, col="red")

## The random function
hist(rEW(n=10000, mu=2, sigma=1.5, nu=0.5), freq=FALSE, 
     xlab="x", las=1, main="")
curve(dEW(x, mu=2, sigma=1.5, nu=0.5), from=0, add=TRUE, col="red") 

## The Hazard function
par(mfrow=c(1,1))
curve(hEW(x, mu=2, sigma=1.5, nu=0.5), from=0, to=2, ylim=c(0, 7), 
      col="red", ylab="Hazard function", las=1)

par(old_par) # restore previous graphical parameters

See Also

EW