dGIW {RelDists}

The Generalized Inverse Weibull distribution

Description

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

Usage

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

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

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

rGIW(n, mu, sigma, nu)

hGIW(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.

Details

The Generalized Inverse Weibull distribution mu, sigma and nu has density given by

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

for x > 0.

Value

dGIW gives the density, pGIW gives the distribution function, qGIW gives the quantile function, rGIW generates random deviates and hGIW gives the hazard function.

Author(s)

Amylkar Urrea Montoya, [e-mail]

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.

Felipe R SdG, Edwin M MO, Gauss M C (2009). “The generalized inverse Weibull distribution.” Statistical papers, 52(3), 591–619. doi:10.1007/s00362-009-0271-3.

Examples

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

## The probability density function
curve(dGIW(x, mu=3, sigma=5, nu=0.5), from=0.001, to=8,
      col="red", ylab="f(x)", las=1)

## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pGIW(x, mu=3, sigma=5, nu=0.5),
      from=0.0001, to=14, col="red", las=1, ylab="F(x)")
curve(pGIW(x, mu=3, sigma=5, nu=0.5, lower.tail=FALSE),
      from=0.0001, to=14, col="red", las=1, ylab="R(x)")

## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qGIW(p, mu=3, sigma=5, nu=0.5), y=p, xlab="Quantile",
     las=1, ylab="Probability")
curve(pGIW(x, mu=3, sigma=5, nu=0.5),
      from=0, add=TRUE, col="red")

## The random function
hist(rGIW(n=1000, mu=3, sigma=5, nu=0.5), freq=FALSE,
     xlab="x", ylim=c(0, 0.8), las=1, main="")
curve(dGIW(x, mu=3, sigma=5, nu=0.5),
      from=0.001, to=14, add=TRUE, col="red")

## The Hazard function
curve(hGIW(x, mu=3, sigma=5, nu=0.5), from=0.001, to=30,
      col="red", ylab="Hazard function", las=1)

par(old_par) # restore previous graphical parameters