A set of generating functions for sufficient statistics for partially observed birth-death process with immigration. The sufficient statistcs are the number of births and immigrations, the mean number of deaths, and the time average of the number of particles.
add.generator(r,s,t,lambda,mu,nu,X0)
rem.generator(r,s,t,lambda,mu,nu,X0)
timeave.laplace(r,s,t,lambda,mu,nu,X0)
hold.generator(w,s,t,lambda,mu,nu,X0)
process.generator(s,time,lambda,mu,nu,X0)
addrem.generator(u, v, s, t, X0, lambda, mu, nu)
remhold.generator( v, w, s, t, X0, lambda, mu, nu)
addhold.generator( u, w, s, t, X0, lambda, mu, nu)
addremhold.generator( u, v, w, s, t, X0, lambda, mu, nu)dummy variable attaining values between 0 and 1. We use r for the single-argument generators and u,v,w for births,deaths, and holdtime for the multi-variable generators syntax, generally.
dummary variable attaining values between 0 and 1
length of the time interval
per particle birth rate
per particle death rate
immigration rate
starting state, a non-negative integer
Numeric value of the corresponding generating function.
Birth-death process is denoted by \(X_t\)
Sufficient statistics are defined as
\(N_t^+\) = number of additions (births and immigrations)
\(N_t^-\) = number of deaths
\(R_t\) = time average of the number of particles, $$\int_0^t X_y dy$$
Function add.generator calculates $$H_i^+(r,s,t) = \sum_{n=0}^\infty \sum_{j=0}^\infty Pr(N_t^+=n,X_t=j | X_o=i) r^n s^j$$
Function rem.generator calculates $$H_i^-(r,s,t) = \sum_{n=0}^\infty \sum_{j=0}^\infty Pr(N_t^-=n,X_t=j | X_o=i) r^n s^j$$
Function timeave.laplace calculates $$H_i^*(r,s,t) = \sum_{j=0}^\infty \int_0^\infty e^{-rx} dPr(R_t \le x, X_t=j | X_o=i) s^j$$
Function processor.generator calculates $$G_i(s,t) = \sum_{j=0}^\infty Pr(X_t=j | X_o=i) r^n s^j$$
Function addrem.generator calculates $$H_i(u,v,s,t) = \sum_{j=0}^\infty \sum_{n_1=0}^\infty \sum_{n_2=0}^\infty Pr(X_t=j, N_t^+=n_1, N_t^-=n_2 | X_o=i) u^{n_1} v^{n_2} s^j$$
Function addhold.generator calculates $$H_i(u,,w,s,t) = \sum_{j=0}^\infty \sum_{n1 \ge 0} u^n_1 \int_0^\infty e^{-rx} dPr(R_t \le x, N_t^+=n_1, X_t=j | X_o=i) s^j$$
Function remhold.generator is the same as addhold.generator but with N- instead of N+.