The Generalized Gompertz distribution

Density, distribution function, quantile function, random generation and hazard function for the generalized Gompertz distribution with parameters mu sigma and nu.

Usage

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

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

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

rGGD(n, mu, sigma, nu)

hGGD(x, mu, sigma, nu)

Arguments

x, q: vector of quantiles.
mu, nu: 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

dGGD gives the density, pGGD gives the distribution function, qGGD gives the quantile function, rGGD generates random deviates and hGGD gives the hazard function.

Details

The Generalized Gompertz Distribution with parameters mu, sigma and nu has density given by

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

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

References

El-Gohary, A., Alshamrani, A., & Al-Otaibi, A. N. (2013). The generalized Gompertz distribution. Applied mathematical modelling, 37(1-2), 13-24.

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 
par(mfrow = c(1, 1))
curve(dGGD(x, mu=1, sigma=0.3, nu=1.5), from = 0, to = 4, 
      col = "red", las = 1, ylab = "f(x)")


## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pGGD(x, mu=1, sigma=0.3, nu=1.5), from = 0, to = 4, 
      ylim = c(0, 1), col = "red", las = 1, ylab = "F(x)")
curve(pGGD(x, mu=1, sigma=0.3, nu=1.5, lower.tail = FALSE), 
      from = 0, to = 4, 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 = qGGD(p=p, mu=1, sigma=0.3, nu=1.5), y = p, 
     xlab = "Quantile", las = 1, ylab = "Probability")
curve(pGGD(x, mu=1, sigma=0.3, nu=1.5), from = 0, add = TRUE, 
      col = "red")

## The random function
hist(rGGD(1000, mu=1, sigma=0.3, nu=1.5), freq = FALSE, xlab = "x", 
     las = 1, ylim = c(0, 0.7), main = "")
curve(dGGD(x,mu=1, sigma=0.3, nu=1.5), from = 0, to =8, add = TRUE, 
      col = "red")


## The Hazard function
par(mfrow=c(1,1))
curve(hGGD(x, mu=1, sigma=0.3, nu=1.5), from = 0, to = 3, col = "red",
      ylab = "The hazard function", las = 1)


par(old_par) # restore previous graphical parameters