# hypoexp: Hypo-Exponential Distribution In sdprisk: Measures of Risk for the Compound Poisson Risk Process with Diffusion

## Description

Density, distribution function, quantile function, random generation and moment-generating function (and its first two derivatives) for the hypo-exponential distribution with rates `rate`.

## Usage

 ```1 2 3 4 5``` ```dhypoexp(x, rate = 1, log = FALSE) phypoexp(q, rate = 1, lower.tail = TRUE, log.p = FALSE, tailarea = FALSE) qhypoexp(p, rate, interval = c(0.0, 1.0e+10)) rhypoexp(n = 1, rate = 1) mgfhypoexp(x, rate = 1, difforder = 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. `difforder` the order of derivative for the moment-generating function; currently only implemented for 0, 1, 2. `rate` vector of (unique) rates. `lower.tail` logical; if `TRUE`, probabilities are P(X ≤ x), otherwise P(X > x). `log, log.p` logical; if `TRUE`, probabilities p are given as log(p). `tailarea` logical; if `TRUE`, probabilities are given for the integrated tail area distribution. `interval` Passed to `uniroot`.

## Details

The sum of n independent exponentially distributed random variables X_{i} with rate parameters λ_{i} has a hypo-exponential distribution with rate vector (λ_{1}, …, λ_{n}).

The hypo-exponential distribution is a generalization of the Erlang distribution (a Gamma distribution with an integer-valued shape parameter) and a special case of the phase-type distribution (see References section).

The quantile function is computed by numeric inversion (using `uniroot`).

## Value

`dhypoexp` gives the density, `phypoexp` gives the distribution function (or the integrated tail area distribution function), `qhypoexp` gives the quantile function, `rhypoexp` generates random deviates and `mgfhypoexp` gives the moment-generating function (or its derivative up to the second order).

## Note

If `length(rate) == 1`, `dhypoexp`, `phypoexp` and `rhypoexp` are equivalent to `dexp`, `pexp` and `rexp` with rate parameter `rate` and should, in fact, be replaced by the latter ones for computation speed.

## References

Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, reprinted and corrected.

`dexp`, `dgamma`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```## Random generation rhypoexp(10, c(3, 5)) ## Mean mu <- mgfhypoexp(0, c(3, 5), difforder = 1) ## Variance mgfhypoexp(0, c(3, 5), difforder = 2) - mu^2 ## Quantile qhypoexp(0.5, c(3, 5)) ```

sdprisk documentation built on May 30, 2017, 2:26 a.m.