dGammaW {RelDists}

The Gamma Weibull distribution

Description

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.

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 \geq 0 and \nu > 0.

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.

Author(s)

Johan David Marin Benjumea, [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.

Stacy EW, others (1962). “A generalization of the gamma distribution.” The Annals of mathematical statistics, 33(3), 1187–1192.

Examples

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

## The probability density function 
curve(dGammaW(x, mu = 0.5, sigma = 2, nu=1), from = 0, to = 6, 
      col = "red", las = 1, ylab = "f(x)")

## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pGammaW(x, mu = 0.5, sigma = 2, nu=1), from = 0, to = 3, 
ylim = c(0, 1), col = "red", las = 1, ylab = "F(x)")
curve(pGammaW(x, mu = 0.5, sigma = 2, nu=1, lower.tail = FALSE), 
from = 0, to = 3, ylim = c(0, 1), col = "red", las = 1, ylab = "R(x)")

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

## The random function
hist(rGammaW(1000, mu = 0.5, sigma = 2, nu=1), freq = FALSE, xlab = "x", 
ylim = c(0, 1), las = 1, main = "")
curve(dGammaW(x, mu = 0.5, sigma = 2, nu=1),  from = 0, add = TRUE, 
col = "red", ylim = c(0, 1))

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

par(old_par) # restore previous graphical parameters