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

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

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

DOBAD documentation built on May 2, 2019, 3:04 a.m.