Description Usage Arguments Details Value Author(s) See Also

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.

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)
``` |

`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 |

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+.

Numeric value of the corresponding generating function.

Marc A. Suchard, Charles Doss

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