Generating functions for birth-death processes with immigration

Description

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.

Usage

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)

Arguments

Details

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) = ∑_{n=0}^∞ ∑_{j=0}^∞ Pr(N_t^+=n,X_t=j | X_o=i) r^n s^j

Function rem.generator calculates

H_i^-(r,s,t) = ∑_{n=0}^∞ ∑_{j=0}^∞ Pr(N_t^-=n,X_t=j | X_o=i) r^n s^j

Function timeave.laplace calculates

H_i^*(r,s,t) = ∑_{j=0}^∞ int_0^∞ e^{-rx} dPr(R_t ≤ x, X_t=j | X_o=i) s^j

Function processor.generator calculates

G_i(s,t) = ∑_{j=0}^∞ Pr(X_t=j | X_o=i) r^n s^j

Function addrem.generator calculates

H_i(u,v,s,t) = ∑_{j=0}^∞ ∑_{n_1=0}^∞ ∑_{n_2=0}^∞ 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) = ∑_{j=0}^∞ ∑_{n1 ≥ 0} u^n_1 int_0^∞ e^{-rx} dPr(R_t ≤ 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+.

Value

Numeric value of the corresponding generating function.

Author(s)

Marc A. Suchard, Charles Doss

See Also

add.joint.mean.many

Results