# pexp: Exponential distribution with piecewise-constant rate In msm: Multi-State Markov and Hidden Markov Models in Continuous Time

## 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

 ```1 2 3 4``` ``` 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 `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), 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 r1,…, rn changing at times t1, …, 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

∏{i=1 … 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.

## 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 [email protected]

`dexp`, `sim.msm`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```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) ```

### Example output

```
```

msm documentation built on Feb. 3, 2018, 1:04 a.m.