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.

- x, q
vector of quantiles.

- shape, rate
vector of shape and rate 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.

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.