pexp {msm} | R Documentation |
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.
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)
x,q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If length(n) > 1 , the length is
taken to be the number required. |
rate |
vector of rates. |
t |
vector of the same length as rate , giving the times at
which the rate changes. The first element of t should be 0,
and t should be in increasing order. |
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]. |
Consider the exponential distribution with rates r1,\dots, rn changing at times t1, \dots, tn, with t1 = 0. Suppose tk is the maximum ti such that ti < x. The density of this distribution at x > 0 is f(x) for k = 1, and
\prod{i=1 \dots k} (1 - F(ti - t{i-1}, r{i-1})) f(x - tk, rk)
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.
dpexp
gives the density, ppexp
gives the distribution
function, qpexp
gives the quantile function, and rpexp
generates random deviates.
C. H. Jackson chris.jackson@mrc-bsu.cam.ac.uk
x <- seq(0.1, 50, by=0.1) rate <- c(0.1, 0.2, 0.05, 0.3) t <- c(0, 10, 20, 30) plot(x, dexp(x, 0.1), type="l") ## standard exponential distribution lines(x, dpexp(x, rate, t), type="l", lty=2) ## distribution with piecewise constant rate plot(x, pexp(x, 0.1), type="l") ## standard exponential distribution lines(x, ppexp(x, rate, t), type="l", lty=2) ## distribution with piecewise constant rate