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(x, prob, rates, log = FALSE)
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 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 <= x], otherwise, P[X > x]. |
order |
order of the moment. |
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^{xX}].
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.
Vincent Goulet vincent.goulet@act.ulaval.ca and Christophe Dutang
http://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)