The Efficiency of Conditional MLE for Pure Birth Processes

The Present paper is devoted to a study of the performance, in large samples, of a conditional maximum likelihood estimator(CMLE) for the parameter ${\lambda}$ in a pure birth processes(PBP). To conduct the conditional inference for the PBP, we drove the likelihood function of time-inhomogeneous Poisson processes. The limiting distributions of CMLE under the likelihoods $L_{t}$ or $\overline{L_{t}}$ are investigated. We found that the CMLE is asymptotically efficient with respect to the both $L_{t}$ or $\overline{L_{t}}$ under the efficiency criterion of Weiss & Wolfowitz(1974).