Density, distribution function, hazards, 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)

hgompertz(x, shape, rate = 1, log = FALSE)

Hgompertz(x, shape, rate = 1, log = FALSE)

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.

`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.

The Gompertz distribution with `shape`

parameter \(a\) and
`rate`

parameter \(b\) has probability density function

$$f(x | a, b) = be^{ax}\exp(-b/a (e^{ax} - 1))$$

and hazard

$$h(x | a, b) = b e^{ax}$$

The hazard is increasing for shape \(a>0\) and decreasing for \(a<0\). 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 \(\exp(b/a)\) of living
forever. On these occasions `qgompertz`

and `rgompertz`

will
return `Inf`

.

Stata Press (2007) Stata release 10 manual: Survival analysis and epidemiological tables.