PhaseType: The Phase-type Distribution

Description

Density, distribution function, random generation, raw moments and moment generating function for the (continuous) Phase-type distribution with parameters prob and rates.

Usage

dphtype(x, prob, rates, log = FALSE)
pphtype(q, prob, rates, lower.tail = TRUE, log.p = FALSE)
rphtype(n, prob, rates)
mphtype(order, prob, rates)
mgfphtype(t, prob, rates, log = FALSE)

Value

dphasetype gives the density,

pphasetype gives the distribution function,

rphasetype generates random deviates,

mphasetype gives the $k$th raw moment, and

mgfphasetype gives the moment generating function in x.

Invalid arguments will result in return value NaN, with a warning.

Arguments

x, q: vector of quantiles.
n: number of observations. If length(n) > 1, the length is taken to be the number required.
prob: vector of initial probabilities for each of the transient states of the underlying Markov chain. The initial probability of the absorbing state is 1 - sum(prob).
rates: square matrix of the rates of transition among the states of the underlying Markov chain.
log, log.p: logical; if TRUE, probabilities/densities $p$ are returned as $\log(p)$.
lower.tail: logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
order: order of the moment.
t: numeric vector.

Author

Vincent Goulet vincent.goulet@act.ulaval.ca and Christophe Dutang

Details

The phase-type distribution with parameters prob $= \pi$ and rates $= \boldsymbol{T}$ has density: $$f(x) = \pi e^{\boldsymbol{T} x} \boldsymbol{t}$$ for $x \ge 0$ and $f(0) = 1 - \pi \boldsymbol{e}$, where $\boldsymbol{e}$ is a column vector with all components equal to one, $\boldsymbol{t} = -\boldsymbol{T} \boldsymbol{e}$ is the exit rates vector and $e^{\boldsymbol{T}x}$ denotes the matrix exponential of $\boldsymbol{T}x$. The matrix exponential of a matrix $\boldsymbol{M}$ is defined as the Taylor series $$e^{\boldsymbol{M}} = \sum_{n = 0}^{\infty} \frac{\boldsymbol{M}^n}{n!}.$$

The parameters of the distribution must satisfy $\pi \boldsymbol{e} \leq 1$, $\boldsymbol{T}_{ii} < 0$, $\boldsymbol{T}_{ij} \geq 0$ and $\boldsymbol{T} \boldsymbol{e} \leq 0$.

The $k$th raw moment of the random variable $X$ is $E[X^k]$ and the moment generating function is $E[e^{tX}]$.

References

https://en.wikipedia.org/wiki/Phase-type_distribution

Neuts, M. F. (1981), Generating random variates from a distribution of phase type, WSC '81: Proceedings of the 13th conference on Winter simulation, IEEE Press.

Examples

Run this code

## Erlang(3, 2) distribution
T <- cbind(c(-2, 0, 0), c(2, -2, 0), c(0, 2, -2))
pi <- c(1,0,0)
x <- 0:10

dphtype(x, pi, T)		# density
dgamma(x, 3, 2)			# same
pphtype(x, pi, T)		# cdf
pgamma(x, 3, 2)			# same

rphtype(10, pi, T)		# random values

mphtype(1, pi, T)		# expected value

curve(mgfphtype(x, pi, T), from = -10, to = 1)

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