The Gamma Weibull distribution

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

Usage

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

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

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

rGammaW(n, mu, sigma, nu)

hGammaW(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

dGammaW gives the density, pGammaW gives the distribution function, qGammaW gives the quantile function, rGammaW generates random deviates and hGammaW gives the hazard function.

Details

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

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

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

References

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

Stacy, E. W. (1962). A generalization of the gamma distribution. The Annals of mathematical statistics, 1187-1192.

Author

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

Examples

# Example 1
# Plotting the mass function for different parameter values

## The probability density function 
curve(dGammaW(x, mu=2, sigma=1.5, nu=0.5), 
      from=0, to=2, 
      col="red", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dGammaW(x, mu=2.4, sigma=1.5, nu=1.3), 
      col="blue", 
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=2.0, sigma=1.5, nu=0.5",
                            "mu=2.4, sigma=1.5, nu=1.3"),
       col=c("red", "blue"), lwd=2, cex=0.6)


# Example 2
# Checking if the cumulative curves converge to 1

curve(pGammaW(x, mu=0.5, sigma=2, nu=1), 
      from=0, to=3, 
      col="red", lwd=2, ylab="F(x)")
curve(pGammaW(x, mu=2.4, sigma=1.5, nu=1.3), 
      col="blue",
      lwd=2,
      add=TRUE)
legend("bottomright", legend=c("mu=2.0, sigma=1.5, nu=0.5",
                               "mu=2.4, sigma=1.5, nu=1.3"),
       col=c("red", "blue"), lwd=2, cex=0.6)


# Example 3
# The quantile function
p <- seq(from=0, to=0.999, length.out=100)
plot(x=qGammaW(p, mu=2.3, sigma=1.7, nu=1.2), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pGammaW(x, mu=2.3, sigma=1.7, nu=1.2), 
      from=0, add=TRUE, col="tomato", lwd=2.5)


# Example 4
# The random function
x <- rGammaW(n=10000, mu=2.4, sigma=1.5, nu=1.3)
hist(x, freq=FALSE)
curve(dGammaW(x, mu=2.4, sigma=1.5, nu=1.3),
      add=TRUE, col="tomato", lwd=2)


# The Hazard function
curve(hGammaW(x, mu=2.4, sigma=1.5, nu=1.3), from=0, to=5, 
      col="red", ylab="Hazard function", las=1)