1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
|
\name{pexp}
\alias{pexp}
\alias{dpexp}
\alias{ppexp}
\alias{qpexp}
\alias{rpexp}
\title{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{
\item{x,q}{vector of quantiles.}
\item{p}{vector of probabilities.}
\item{n}{number of observations. If \code{length(n) > 1}, the length is
taken to be the number required.}
\item{rate}{vector of rates.}
\item{t}{vector of the same length as \code{rate}, giving the times at
which the rate changes. The first element of \code{t} should be 0,
and \code{t} should be in increasing order.}
\item{log, log.p}{logical; if TRUE, probabilities p are given as
log(p).}
\item{lower.tail}{logical; if TRUE (default), probabilities are P[X <= x],
otherwise, P[X > x].}
}
\value{
\code{dpexp} gives the density, \code{ppexp} gives the distribution
function, \code{qpexp} gives the quantile function, and \code{rpexp}
generates random deviates.
}
\details{
Consider the exponential distribution with rates \eqn{r_1, \ldots,
r_n}{r1,\dots, rn} changing at times \eqn{t_1, \ldots, t_n}{t1,
\dots, tn}, with \eqn{t_1 = 0}{t1 = 0}. Suppose \eqn{t_k}{tk} is the
maximum \eqn{t_i}{ti} such that
\eqn{t_i < x}{ti < x}.
The density of this distribution at \eqn{x > 0} is \eqn{f(x)} for \eqn{k = 1},
and \deqn{\prod_{i=1}^k (1 - F(t_{i} - t_{i-1}, r_i)) f(x - t_{k}, r_{k})}{\prod{i=1 \dots k} (1 - F(ti - t{i-1}, r{i-1})) f(x - tk, rk)}
for k > 1.
where \eqn{F()} and \eqn{f()} are the distribution and density
functions of the standard exponential distribution.
If \code{rate} is of length 1, this is just the standard exponential
distribution. Therefore, for example, \code{dpexp(x)}, with no other
arguments, is simply equivalent to \code{dexp(x)}.
Only \code{rpexp} is used in the \code{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.
}
\seealso{
\code{\link{dexp}}, \code{\link{sim.msm}}.
}
\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)
}
\author{C. H. Jackson \email{chris.jackson@mrc-bsu.cam.ac.uk}}
\keyword{distribution}
|
