Description Usage Arguments Details Value Note Author(s) References See Also
Density, distribution function, hazards, quantile function and random generation for the Gompertz distribution with unrestricted shape.
1 2 3 4 5 6 |
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
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 <= x), otherwise, P(X > x). |
The Gompertz distribution with shape
parameter a and
rate
parameter b has probability density function
f(x | a, b) = b exp(ax) exp(-b/a (exp(ax) - 1))
and hazard
h(x | a, b) = b exp(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 (exp(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
.
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.
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 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.
Christopher Jackson <chris.jackson@mrc-bsu.cam.ac.uk>
Stata Press (2007) Stata release 10 manual: Survival analysis and epidemiological tables.
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