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

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  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)

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].

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 lambda1) state 1 to state 3 (intensity mu1) and state 2 to state 3 (intensity mu2). States 1 and 2 are the two "phases" and state 3 is the "exit" state.

The density is

f(t | l1, mu1) = exp(-(l1+mu1)*t)*(mu1 + (l1+mu1)*l1*t)

if l1 + mu1 = mu2, and

f(t | l1, mu1, mu2) = ((l1+mu1)*exp(-(l1+mu1)*t)*(mu2-mu1) + mu2*l1*exp(-mu2*t))/(l1+mu1-mu2)

otherwise. The distribution function is

F(t | l1, mu1) = 1 - exp(-(l1+mu1)*t)*(1 + l1*t)

if l1 + mu1 = mu2, and

F(t | l1, mu1, mu2) = 1 - (exp(-(l1+mu1)*t)*(-mu1+mu2) + l1*exp(-mu2*t))/(l1+mu1-mu2)

otherwise. Quantiles are calculated by numerically inverting the distribution function.

The mean is (1 + l1/mu2) / (l1 + mu1).

The variance is (2 + 2*l1*(l1+mu1+ mu2)/mu2^2 - (1 + l1/mu2)^2)/(l1+mu1)^2.

If mu1=mu2 it reduces to an exponential distribution with rate mu1, and the parameter l1 is redundant. Or also if l1=0.

The hazard at x=0 is μ_1, and smoothly increasing if mu1<mu2. If l1 + mu1 >= mu2 it increases to an asymptote of mu2, and if l1 + mu1 <= mu2 it increases to an asymptote of l1 + mu1. The hazard is decreasing if mu1>mu2, to an asymptote of mu2.

Value

d2phase gives the density, p2phase gives the distribution function, q2phase gives the quantile function, r2phase generates random deviates, and h2phase gives the hazard.

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 M1 = 1/(l1+mu1) and sojourn distribution is exponential with rate l1+mu1.

2) "long stayers" whose mean sojourn time 1/(l1+mu1) + 1/mu2 and sojourn distribution is the sum of two exponentials with rate l1+mu1 and mu2 respectively. The individual is a "long stayer" with probability p=λ_1/(λ_1 + μ_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 l1=p/M1, mu1=(1-p)/M1, and mu2 = 1/(M2 - M1). 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 M2 < 2M1, and also only if (M2 - 2M1)/(M2 - M1) < p.

For increasing hazards with l1 + mu1 <= mu2, the maximum hazard ratio between any time t and time 0 is 1/(1-p).

For increasing hazards with l1 + mu1 >= mu2, the maximum hazard ratio is M1/((1-p)(M2 - M1)). 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(s)

C. H. Jackson chris.jackson@mrc-bsu.cam.ac.uk

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

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