msm (version 1.7)

2phase: Coxian phase-type distribution with two phases

Description

Density, distribution, quantile functions and other utilities for the Coxian phase-type distribution with two phases.

Usage

d2phase(x, l1, mu1, mu2, log=FALSE)
  p2phase(q, l1, mu1, mu2, lower.tail=TRUE, log.p=FALSE)
  q2phase(p, l1, mu1, mu2, lower.tail=TRUE, log.p=FALSE)
  r2phase(n, l1, mu1, mu2)
  h2phase(x, l1, mu1, mu2, log=FALSE)

Value

d2phase gives the density, p2phase gives the distribution function, q2phase gives the quantile function, r2phase

generates random deviates, and h2phase gives the hazard.

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.

l1

Intensity for transition between phase 1 and phase 2.

mu1

Intensity for transition from phase 1 to exit.

mu2

Intensity for transition from phase 2 to exit.

log

logical; if TRUE, return log density or log hazard.

log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

Alternative parameterisation

An individual following this distribution can be seen as coming from a mixture of two populations:

1) "short stayers" whose mean sojourn time is $M_1=1/({\lambda }_1+{\mu }_1)$ and sojourn distribution is exponential with rate ${\lambda }_1+{\mu }_1$.

2) "long stayers" whose mean sojourn time $M_2=1/({\lambda }_1+{\mu }_1)+1/{\mu }_2$ and sojourn distribution is the sum of two exponentials with rate ${\lambda }_1+{\mu }_1$ and ${\mu }_2$ respectively. The individual is a "long stayer" with probability $p={\lambda }_1/({\lambda }_1+{\mu }_1)$.

Thus a two-phase distribution can be more intuitively parameterised by the short and long stay means $M_1<M_2$ and the long stay probability $p$. Given these parameters, the transition intensities are ${\lambda }_1=p/M_1$, ${\mu }_1=(1-p)/M_1$, and ${\mu }_2=1/(M_2-M_1)$. This can be useful for choosing intuitively reasonable initial values for procedures to fit these models to data.

The hazard is increasing at least if $M_2<2M_1$, and also only if $(M_2-2M_1)/(M_2-M_1)<p$.

For increasing hazards with ${\lambda }_1+{\mu }_1\le {\mu }_2$, the maximum hazard ratio between any time $t$ and time 0 is $1/(1-p)$.

For increasing hazards with ${\lambda }_1+{\mu }_1\ge {\mu }_2$, the maximum hazard ratio is $M_1/((1-p)(M_2-M_1))$. This is the minimum hazard ratio for decreasing hazards.

General phase-type distributions

This is a special case of the n-phase Coxian phase-type distribution, which in turn is a special case of the (general) phase-type distribution. The actuar R package implements a general n-phase distribution defined by the time to absorption of a general continuous-time Markov chain with a single absorbing state, where the process starts in one of the transient states with a given probability.

Author

C. H. Jackson

Details

This is the distribution of the time to reach state 3 in a continuous-time Markov model with three states and transitions permitted from state 1 to state 2 (with intensity ${\lambda }_1$) state 1 to state 3 (intensity ${\mu }_1$) and state 2 to state 3 (intensity ${\mu }_2$). States 1 and 2 are the two "phases" and state 3 is the "exit" state.

The density is

$$f(t|{\lambda }_1,{\mu }_1)=e^{-({\lambda }_1+{\mu }_1)t}({\mu }_1+({\lambda }_1+{\mu }_1){\lambda }_{1}t)$$

if ${\lambda }_1+{\mu }_1={\mu }_2$, and

$$f(t|{\lambda }_1,{\mu }_1,{\mu }_2)=\frac{({\lambda }_1+{\mu }_1)e^{-({\lambda }_1+{\mu }_1)t}({\mu }_2-{\mu }_1)+{\mu }_2{\lambda }_1{e}^{-{\mu }_{2}t}}{{\lambda }_1+{\mu }_1-{\mu }_2}$$

otherwise. The distribution function is

$$F(t|{\lambda }_1,{\mu }_1)=1-e^{-({\lambda }_1+{\mu }_1)t}(1+{\lambda }_{1}t)$$

if ${\lambda }_1+{\mu }_1={\mu }_2$, and

$$F(t|{\lambda }_1,{\mu }_1,{\mu }_2)=1-\frac{e^{-({\lambda }_1+{\mu }_1)t}({\mu }_2-{\mu }_1)+{\lambda }_1{e}^{-{\mu }_{2}t}}{{\lambda }_1+{\mu }_1-{\mu }_2}$$ otherwise. Quantiles are calculated by numerically inverting the distribution function.

The mean is $(1+{\lambda }_1/{\mu }_2)/({\lambda }_1+{\mu }_1)$.

The variance is $(2+2{\lambda }_1({\lambda }_1+{\mu }_1+{\mu }_2)/{\mu }_2^2-(1+{\lambda }_1/{\mu }_2{)}^2)/({\lambda }_1+{\mu }_1{)}^2$.

If ${\mu }_1={\mu }_2$ it reduces to an exponential distribution with rate ${\mu }_1$, and the parameter ${\lambda }_1$ is redundant. Or also if ${\lambda }_1=0$.

The hazard at $x=0$ is ${\mu }_1$, and smoothly increasing if ${\mu }_1<{\mu }_2$. If ${\lambda }_1+{\mu }_1\ge {\mu }_2$ it increases to an asymptote of ${\mu }_2$, and if ${\lambda }_1+{\mu }_1\le {\mu }_2$ it increases to an asymptote of ${\lambda }_1+{\mu }_1$. The hazard is decreasing if ${\mu }_1>{\mu }_2$, to an asymptote of ${\mu }_2$.

References

C. Dutang, V. Goulet and M. Pigeon (2008). actuar: An R Package for Actuarial Science. Journal of Statistical Software, vol. 25, no. 7, 1-37. URL http://www.jstatsoft.org/v25/i07