The present paper is devoted to a study of the performance, in large samples, of a conditional maximum likelihood estimator for a parameter in a pure birth processes.
In chapter 2, we have deriven the probability density function of the sojourn times and the likelihood function for the pure birth processes with birth rate λ and initial state q.
In chapter 3, the efficiency criterions for the case of IID, not IID, and the case of having different limiting distributions of the estimators, are introduced. The criterions are redefined for the models of continuous time stochastic processes.
We introduce the concepts of mixture experiment and conditional inferences in section 4.1. To conduct the conditional inference for the pure birth processes, we have deriven the likelihood function of time-inhomogeneous Poisson processes and we have confirmed that the likelihood of the Poisson processes is ergodic model. The conditional likelihood and mixture likelihood are computed to derive likelihoods which contain only a parameter of interest. The limiting listributions of conditional maiximum likelihood estimator under the likelihoods Lt or Lt are deriverd in the lemma. We found that the conditional maximum likelihood estimator is asymptotically efficient with respect to the both the reference likelihoods Lt or Lt if we use the efficiency criterion of Weiss & Wolfowitz(1974).
Thus, we may conclude that the conditional likelihood Ltc is fully informative for point estimation, regardless of whether one chooses to use Lt or Lt as the reference model. The conditionality principle here is fully validated both logical efficiency grounds.