The four parameter Exponentiated Generalized Gamma distribution

Density, distribution function, quantile function, random generation and hazard function for the four parameter Exponentiated Generalized Gamma distribution with parameters mu, sigma, nu and tau.

Usage

dEGG(x, mu, sigma, nu, tau, log = FALSE)

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

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

rEGG(n, mu, sigma, nu, tau)

hEGG(x, mu, sigma, nu, tau)

Arguments

x, q

vector of quantiles.

mu

parameter.

sigma

parameter.

nu

parameter.

tau

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

dEGG gives the density, pEGG gives the distribution function, qEGG gives the quantile function, rEGG generates random deviates and hEGG gives the hazard function.

Details

Four-Parameter Exponentiated Generalized Gamma distribution with parameters mu, sigma, nu and tau has density given by

\(f(x) = \frac{\nu \sigma}{\mu \Gamma(\tau)} \left(\frac{x}{\mu}\right)^{\sigma \tau -1} \exp\left\{ - \left( \frac{x}{\mu} \right)^\sigma \right\} \left\{ \gamma_1\left( \tau, \left( \frac{x}{\mu} \right)^\sigma \right) \right\}^{\nu-1} ,\)

for \(x > 0\).

References

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

Cordeiro, G. M., Ortega, E. M., & Silva, G. O. (2011). The exponentiated generalized gamma distribution with application to lifetime data. Journal of statistical computation and simulation, 81(7), 827-842.

See also

EGG

Author

Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co

Examples

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

## The probability density function
curve(dEGG(x, mu=0.1, sigma=0.8, nu=10, tau=1.5), from=0.000001, to=1.5, ylim=c(0, 2.5),
      col="red", las=1, ylab="f(x)")


## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pEGG(x, mu=0.1, sigma=0.8, nu=10, tau=1.5),
      from=0.000001, to=1.5, col="red", las=1, ylab="F(x)")
curve(pEGG(x, mu=0.1, sigma=0.8, nu=10, tau=1.5, lower.tail=FALSE),
      from=0.000001, to=1.5, col="red", las=1, ylab="R(x)")


## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qEGG(p, mu=0.1, sigma=0.8, nu=10, tau=1.5), y=p, xlab="Quantile",
     las=1, ylab="Probability")
curve(pEGG(x, mu=0.1, sigma=0.8, nu=10, tau=1.5), 
      from=0.00001, add=TRUE, col="red")

## The random function
hist(rEGG(n=100, mu=0.1, sigma=0.8, nu=10, tau=1.5), freq=FALSE,
     xlab="x", las=1, main="")
curve(dEGG(x, mu=0.1, sigma=0.8, nu=10, tau=1.5),
      from=0.0001, to=2, add=TRUE, col="red")


## The Hazard function
curve(hEGG(x,  mu=0.1, sigma=0.8, nu=10, tau=1.5), from=0.0001, to=1.5,
      col="red", ylab="Hazard function", las=1)

par(old_par) # restore previous graphical parameters