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

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"))

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).

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.

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.