PhaseType {actuar} | R Documentation |

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

and `rates`

.

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)

`x, q` |
vector of quantiles. |

`n` |
number of observations. If |

`prob` |
vector of initial probabilities for each of the transient
states of the underlying Markov chain. The initial probability of
the absorbing state is |

`rates` |
square matrix of the rates of transition among the states of the underlying Markov chain. |

`log, log.p` |
logical; if |

`lower.tail` |
logical; if |

`order` |
order of the moment. |

`t` |
numeric vector. |

The phase-type distribution with parameters `prob`

*=
pi* and `rates`

*= T* has density:

*
f(x) = pi %*% exp(T * x) %*% t*

for *x ≥ 0* and *f(0) = 1 - pi
%*% e*, where
*e*
is a column vector with all components equal to one,
*
t = -T %*% e*
is the exit rates vector and
*exp(T * x)*
denotes the matrix exponential of *T * x*. The
matrix exponential of a matrix *M* is defined as
the Taylor series

*
exp(M) = sum(n = 0:Inf; (M^n)/(n!)).*

The parameters of the distribution must satisfy
*pi %*% e <= 1*,
*T[i, i] < 0*,
*T[i, j] >= 0* and
*T %*% e <= 0*.

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

`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.

The `"distributions"`

package vignette provides the
interrelations between the continuous size distributions in
actuar and the complete formulas underlying the above functions.

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

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.

## 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)

[Package *actuar* version 3.1-4 Index]