extraDistr (version 1.9.1)

Gompertz: Gompertz distribution

Description

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

Usage

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 \(P[X \le x]\) otherwise, \(P[X > x]\).

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 $$ f(x) = a \exp\left(bx - \frac{a}{b} (\exp(bx)-1)\right) $$

Cumulative distribution function $$ F(x) = 1-\exp\left(-\frac{a}{b} (\exp(bx)-1)\right) $$

Quantile function $$ F^{-1}(p) = \frac{1}{b} \log\left(1-\frac{b}{a}\log(1-p)\right) $$

References

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

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)