Gompertz: The Gompertz distribution

Description

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

Arguments

x, q

vector of quantiles.

vector of probabilities.

number of observations. If length(n) > 1, the length is taken to be the number required.

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, $P(X > x)$.

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="" exponential="" distribution="" with="" constant="" hazard="" and="" rate="" $b$.="" probability="" function="" $$f(x="" |="" a,="" b)="1" -="" \exp(-b="" a="" (e^{ax}="" 1))$$="" thus="" if="" $a$="" negative,="" letting="" $x$="" tend="" infinity="" shows="" that="" there="" non-zero="" $1="" \exp(b="" a)$="" of="" living="" forever.="" on="" these="" occasions="" qgompertz and rgompertz will return Inf.

References