Efficient sequential estimation in continuous time branching processes and countintg processes

Abstract

This thesis is concerned with the efficient sequential estimation in continuous time branching processes and in multivariate counting processes. Determining a sequential estimation scheme involves the problem of defining and then finding optimal stopping rules or sampling plans. A lower bound on the variance of an unbiased estimator is given by the (fixed time or sequential) Cramer-Rao type information inequality. If, under a sampling plan S, the equality is attained for an estimator f of a parametric function g = E(f), then S and f are said to be ``efficient`` and g is said to be ``efficiently estimable``. All efficient triples (S,f,g) in a continuous time branching process with immigration with split rate $\lambda_1$, immigration rate $\lambda_2$, offspring distribution [$p_{1j}$, $j\geq0$, $j\neq1$] and immigration distribution [$p_{2j}$, $j\geq1$] are characterized: it is shown that only linear combinations of $\lambda_ip_{ij}$``s and ratios of linear combinations of $\lambda_i^{-1}$ and $\lambda_ip_{ij}$``s to the partial sums of $p_{ij}$``s, i = 1,2, are efficiently estimable when the offspring and immigration distributions are assumed to have finite supports. It is also shown that only the linear combinations of $\mu_i^{-1}$ and $(\lambda_i\mu_i)^{-1}$, the linear combinations of $\mu_i^{-1}$ and $(\lambda_i\mu_i)^{-1}$, and the linear combinations of $\lambda_i^{-1}$ and $\mu_i$ are efficiently estimable when [$p_{ij}$] is of power series type, i = 1,2, where $\mu_1$ and $\mu_2$ are the offspring and immigration means, respectively. Applications to some simple models such as a birth process, a linear birth and death process with immigration and an M/M/$\infty$ queue are also discussed. Efficient sequential estimation in a continuous time branching process without immigration is also considered. Because of the possibility of early, extinction, however, only conditional analysis is considered given either a large initial population or on the set of nonext...