fj_xmax_coxian.h File Reference

Expected maximum of K i.i.d. More...

#include "line/api/fj/fj_types.h"
#include "line/num/number.h"
#include "line/util/error.h"

Include dependency graph for fj_xmax_coxian.h:

Go to the source code of this file.

Classes
struct	line::fj::FJXmaxCoxianResult< T >
	[Xmax, m1, c2] of fj_xmax_coxian. More...

Namespaces
namespace	line
namespace	line::fj

Functions
template<class T>
FJXmaxCoxianResult< T >	line::fj::fj_xmax_coxian (unsigned K, const T &mu1, const T &mu2, const T &q)
	Expected maximum of K i.i.d.

Detailed Description

Expected maximum of K i.i.d.

two-stage Coxian variables.

Templated port of matlab/src/api/fj/fj_xmax_coxian.m.

With X = T1 + B T2, T1 ~ Exp(mu1), T2 ~ Exp(mu2) and B ~ Bernoulli(q), the survival function is a two-term exponential mixture

S(t) = A exp(-mu1 t) + B exp(-mu2 t), A = (1-q) + q mu2/(mu2-mu1), B = -q mu1/(mu2-mu1),

so expanding 1 - (1-S)^K binomially and integrating term by term gives

E[Y_K] = sum_{j=1..K} (-1)^(j+1) C(K,j) sum_{i=0..j} C(j,i) A^(j-i) B^i / ((j-i) mu1 + i mu2).

At coincident stage rates the mixture degenerates into S(t) = (1 + q mu t) exp(-mu t) and the same expansion is carried out with integral t^m exp(-j mu t) dt = m!/(j mu)^(m+1), which is selected automatically.

Definition in file fj_xmax_coxian.h.