Gompertz function

Gompertz distribution

Gompertz distribution

Density, distribution function, quantile function and random generation for the Gompertz distribution.

dgompertz(x, a = 1, b = 1, log = FALSE)

pgompertz(q, a = 1, b = 1, lower.tail = TRUE, log.p = FALSE)

qgompertz(p, a = 1, b = 1, lower.tail = TRUE, log.p = FALSE)

rgompertz(n, a = 1, b = 1)

Arguments

  • x, q: vector of quantiles.

  • a, b: positive valued scale and location 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]P[X \le x]

    otherwise, P[X>x]P[X > x].

  • p: vector of probabilities.

  • n: number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)=aexp⁡(bx−ab(exp⁡(bx)−1))f(x)=a∗exp(b∗x−a/b∗(exp(b∗x)−1)) f(x) = a \exp\left(bx - \frac{a}{b} (\exp(bx)-1)\right)f(x) = a*exp(b*x - a/b * (exp(b*x)-1))

Cumulative distribution function

F(x)=1−exp⁡(−ab(exp⁡(bx)−1))F(x)=1−exp(−a/b∗(exp(b∗x)−1)) F(x) = 1-\exp\left(-\frac{a}{b} (\exp(bx)-1)\right)F(x) = 1-exp(-a/b * (exp(b*x)-1))

Quantile function

F−1(p)=1blog⁡(1−balog⁡(1−p))F−1(p)=1/b∗log(1−b/a∗log(1−p)) F^{-1}(p) = \frac{1}{b} \log\left(1-\frac{b}{a}\log(1-p)\right)F^-1(p) = 1/b * log(1 - b/a * log(1-p))

Examples

x <- rgompertz(1e5, 5, 2)
hist(x, 100, freq = FALSE)
curve(dgompertz(x, 5, 2), 0, 1, col = "red", add = TRUE)
hist(pgompertz(x, 5, 2))
plot(ecdf(x))
curve(pgompertz(x, 5, 2), 0, 1, col = "red", lwd = 2, add = TRUE)

References

Lenart, A. (2012). The Gompertz distribution and Maximum Likelihood Estimation of its parameters - a revision. MPIDR WORKING PAPER WP 2012-008. https://www.demogr.mpg.de/papers/working/wp-2012-008.pdf

extraDistr packageRead PDF manual
  • Maintainer: Tymoteusz Wolodzko
  • License: GPL-2
  • Last published: 2023-11-30

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