From Species Problem to a General Coverage Problem via a New Interpretation

摘要

A new interpretation is given, which provides another way of understanding the structure of the species problem and sheds light on the properties of a general coverage problem. As an illustrative example, the popular Turing-Good-Robbins estimator is shown to be a natural choice from this interpretation in the species problem. We set up a general framework of various coverage problems in this paper. The new interpretation is applied to this general situation which leads to many interesting applications in addition to the species problem. The coverage problems considered in this paper include the species problem, the problem of estimating the volume of a convex set, and the missile-coverage problem. It is pointed out that the general estimators derived from this new interpretation usually estimate the probabilistic phenomenon involving only n - 1 observations which may not be appropriate. A general modified procedure is thus suggested to improve the current estimators. To justify the interpretation theoretically, we present some limit theorems in terms of species problems, even though the results are expected to hold more generally. (kr)