Density, distribution function, hazards, quantile function and random generation for the Gompertz distribution with unrestricted shape.
Usage
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)Arguments
- 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.
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.
Details
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.
Note
Some implementations of the Gompertz restrict \(a\) to be strictly
positive, which ensures that the probability of survival decreases to zero
as \(x\) increases to infinity. The more flexible implementation given
here is consistent with streg in Stata.
The functions eha::dgompertz and similar available in the
package eha label the parameters the other way round, so that what is
called the shape there is called the rate here, and what is
called 1 / scale there is called the shape here. The
terminology here is consistent with the exponential dexp and
Weibull dweibull distributions in R.
References
Stata Press (2007) Stata release 10 manual: Survival analysis and epidemiological tables.