Gompertz: The Gompertz Distribution

Description

Density, distribution function, quantile function, hazard function, cumulative hazard function, and random generation for the Gompertz distribution with parameters shape and scale.

Usage

dgompertz(x, shape = 1, scale = 1, rate, log = FALSE, 
param = c("default", "canonical", "rate")) 
pgompertz(q, shape = 1, scale = 1, rate, lower.tail = TRUE, log.p = FALSE, 
param = c("default", "canonical", "rate")) 
qgompertz(p, shape = 1, scale = 1, rate, lower.tail = TRUE, log.p = FALSE, 
param = c("default", "canonical", "rate")) 
hgompertz(x, shape = 1, scale = 1, rate, log = FALSE, 
param = c("default", "canonical", "rate")) 
Hgompertz(x, shape = 1, scale = 1, rate, log.p = FALSE, 
param = c("default", "canonical", "rate")) 
rgompertz(n, shape = 1, scale = 1, rate, 
param = c("default", "canonical", "rate"))

Value

dgompertz gives the density, pgompertz gives the distribution function, qgompertz gives the quantile function, hgompertz gives the hazard function, Hgompertz gives the cumulative hazard function, and rgompertz generates random deviates.

Invalid arguments will result in return value NaN, with a warning.

Arguments

shape, scale: shape and scale parameters, both defaulting to 1.
rate: the rate parameter for that parametrization, replaces scale.
lower.tail: logical; if TRUE (default), probabilities are $P(X \le x)$, otherwise, $P(X > x)$.
param: default or canonical or rate.
x, q: vector of quantiles.
p: vector of probabilities.
n: number of observations. If length(n) > 1, the length is taken to be the number required.
log, log.p: logical; if TRUE, probabilities p are given as log(p).

Details

The Gompertz distribution with scale parameter $a$ and shape parameter $\sigma$ has hazard function given by $$h(x) = a \exp(x/\sigma)$$ for $x \ge 0$. If param = "canonical", then then a --> a/b, so that b is a true scale parameter (for any fixed a), and b is an 'AFT parameter'. If param = "rate", then b --> 1/b.