Density, distribution function, quantile function and random generation for the Gompertz distribution with unrestricted shape.
dGompertz(x, shape, rate = 1, log = FALSE)
pGompertz(q, shape, rate = 1, lower.tail = TRUE, log.p = FALSE)
qGompertz(p, shape, rate = 1, lower.tail = TRUE, log.p = FALSE)
rGompertz(n, shape = 1, rate = 1)dGompertz gives the density, pGompertz gives the
distribution function, qGompertz gives the quantile function,
and rGompertz generates random deviates.
vector of quantiles.
vector of shape and rate parameters.
logical; if TRUE, probabilities p are given as log(p).
logical; if TRUE (default), probabilities are \(P(X \le x)\), otherwise, \(P(X > x)\).
vector of probabilities.
number of observations. If length(n) > 1, the length is
taken to be the number required.
Christopher Jackson <chris.jackson@mrc-bsu.cam.ac.uk>
The Gompertz distribution with shape parameter \(a\) and
rate parameter \(b\) has probability density function
$$f(x | a, b) = be^{ax}\exp(-b/a (e^{ax} - 1))$$
For \(a=0\) the Gompertz is equivalent to the exponential distribution with constant hazard and rate \(b\).
The probability distribution function is $$F(x | a, b) = 1 - \exp(-b/a (e^{ax} - 1))$$
Thus if \(a\) is negative, letting \(x\) tend to infinity shows that
there is a non-zero probability \(1 - \exp(b/a)\) of living
forever. On these occasions qGompertz and rGompertz will
return Inf.
Stata Press (2007) Stata release 10 manual: Survival analysis and epidemiological tables.