The Phase-type Distribution
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 kth 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 kth 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
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)
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