pexp {msm} | R Documentation |
Exponential distribution with piecewise-constant rate
Description
Density, distribution function, quantile function and random generation for a generalisation of the exponential distribution, in which the rate changes at a series of times.
Usage
dpexp(x, rate=1, t=0, log = FALSE)
ppexp(q, rate=1, t=0, lower.tail = TRUE, log.p = FALSE)
qpexp(p, rate=1, t=0, lower.tail = TRUE, log.p = FALSE)
rpexp(n, rate=1, t=0)
Arguments
x , q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
rate |
vector of rates. |
t |
vector of the same length as |
log , log.p |
logical; if TRUE, probabilities p are given as log(p), or log density is returned. |
lower.tail |
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]. |
Details
Consider the exponential distribution with rates r_1, \ldots,
r_n
changing at times t_1, \ldots, t_n
, with t_1 = 0
. Suppose t_k
is the
maximum t_i
such that
t_i < x
.
The density of this distribution at x > 0
is f(x)
for k = 1
,
and
\prod_{i=1}^k (1 - F(t_{i} - t_{i-1}, r_i)) f(x - t_{k}, r_{k})
for k > 1.
where F()
and f()
are the distribution and density
functions of the standard exponential distribution.
If rate
is of length 1, this is just the standard exponential
distribution. Therefore, for example, dpexp(x)
, with no other
arguments, is simply equivalent to dexp(x)
.
Only rpexp
is used in the msm
package, to simulate
from Markov processes with piecewise-constant intensities depending on
time-dependent covariates. These functions are merely provided for
completion, and are not optimized for numerical stability or speed.
Value
dpexp
gives the density, ppexp
gives the distribution
function, qpexp
gives the quantile function, and rpexp
generates random deviates.
Author(s)
C. H. Jackson chris.jackson@mrc-bsu.cam.ac.uk
See Also
Examples
x <- seq(0.1, 50, by=0.1)
rate <- c(0.1, 0.2, 0.05, 0.3)
t <- c(0, 10, 20, 30)
## standard exponential distribution
plot(x, dexp(x, 0.1), type="l")
## distribution with piecewise constant rate
lines(x, dpexp(x, rate, t), type="l", lty=2)
## standard exponential distribution
plot(x, pexp(x, 0.1), type="l")
## distribution with piecewise constant rate
lines(x, ppexp(x, rate, t), type="l", lty=2)