dOW function

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.

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.

Returns

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

c("f(x) = \\left( \\frac{\\sigma\\nu}{x} \\right) (\\mu x)^\\sigma\n", " e^{(\\mu x)^\\sigma} \\left(e^{(\\mu x)^{\\sigma}}-1\\right)^{\\nu-1}\n", " \\left[ 1 + \\left(e^{(\\mu x)^{\\sigma}}-1\\right)^\\nu \\right]^{-2}")

for x > 0.

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

References

Rdpack::insert_ref(key="Cooray2006",package="RelDists")

Author(s)

Jaime Mosquera Gutiérrez jmosquerag@unal.edu.co