Initial size estimation for the linear pure death process

SUMMARY Consider a linear pure death process in which the common death rate and the initial size n of the population are both unknown. An estimate of n is required, based on the observed times of the firstj deaths, i.e. with censored sampling. It is shown that the only reliable non- Bayesian method is to use interval estimates, for all the standard point estimation techniques are liable to fail. Appropriate interval estimators are derived and discussed. The corresponding truncated sampling model is briefly considered. The linear pure death process may be characterized as representing a dying population in which the lifetimes are independently and identically distributed, each with probability density Ae - t(t > 0), for some A > 0. We assume that both the initial population size n and the rate parameter A are unknown, and our aim is to estimate n. For many years work on the estimation of population size, such as by Johnson (1962), Hoel (1968) and Marcus & Blumenthal (1974), assumed a lifetime distribution which was general, but completely specified. Recently interest has been shown in lifetime distributions with unknown parameters, in particular in papers by Sanathanan (1977), who gives asymptotic results for maximum likelihood estimation, and Blumenthal & Marcus (1975), who consider precisely the above problem. We consider mainly the censored sampling model, under which an estimate of n is required immediately after the jth death, where j is a preassigned integer known not to exceed n. In ? 2 we derive an invariantly sufficient statistic and its distribution. In ? 3 we discuss point estimation of n: unbiased estimators do not exist, while the methods of moments and of maximum likelihood are both liable to fail in that there is positive probability, sometimes close to 2, that the equation for the estimator has no finite solution. In ? 4 we derive an interval estimator and discuss its properties. While an interval estimate is always available, it is in some circumstances quite vague. This brings out clearly that for the present problem it is more than usually inadvisable to rely on point