The Inverse Weibull distribution

Density, distribution function, quantile function, random generation and hazard function for the inverse weibull distribution with parameters mu and sigma.

Usage

dIW(x, mu, sigma, log = FALSE)

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

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

rIW(n, mu, sigma)

hIW(x, mu, sigma)

Arguments

x, q: vector of quantiles.
mu: scale parameter.
sigma: 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.

Value

dIW gives the density, pIW gives the distribution function, qIW gives the quantile function, rIW generates random deviates and hIW gives the hazard function.

Details

The inverse weibull distribution with parameters mu and sigma has density given by

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

for \(x > 0\), \(\mu > 0\) and \(\sigma > 0\)

References

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

Drapella, A. (1993). The complementary Weibull distribution: unknown or just forgotten?. Quality and reliability engineering international, 9(4), 383-385.

Author

Johan David Marin Benjumea, johand.marin@udea.edu.co

Examples

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

## The probability density function
curve(dIW(x, mu=5, sigma=2.5), from=0, to=10,
      ylim=c(0, 0.55), col="red", las=1, ylab="f(x)")

#'
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pIW(x, mu=5, sigma=2.5),
      from=0, to=10,  col="red", las=1, ylab="F(x)")
curve(pIW(x, mu=5, sigma=2.5, lower.tail=FALSE),
      from=0, to=10, col="red", las=1, ylab="R(x)")

            
## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qIW(p, mu=5, sigma=2.5), y=p, xlab="Quantile",
  las=1, ylab="Probability")
curve(pIW(x, mu=5, sigma=2.5), from=0, add=TRUE, col="red")
  
## The random function
hist(rIW(n=10000, mu=5, sigma=2.5), freq=FALSE, xlim=c(0,60),
  xlab="x", las=1, main="")
curve(dIW(x, mu=5, sigma=2.5), from=0, add=TRUE, col="red")


## The Hazard function
par(mfrow=c(1,1))
curve(hIW(x, mu=5, sigma=2.5), from=0, to=15, ylim=c(0, 0.9),
   col="red", ylab="Hazard function", las=1)


par(old_par) # restore previous graphical parameters