# add.generator: Generating functions for birth-death processes with... In DOBAD: Analysis of Discretely Observed Linear Birth-and-Death(-and-Immigration) Markov Chains

## 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

 1 2 3 4 5 6 7 8 9 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

 r,u,v,w 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. s dummary variable attaining values between 0 and 1 t,time length of the time interval lambda per particle birth rate mu per particle death rate nu immigration rate X0 starting state, a non-negative integer

## 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

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

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

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

add.joint.mean.many