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

Abstract

A basic interpretation is given which provides a new way of understanding the structure of the species problem and which leads to the popular Turing-Good-Robbins estimator. Through this interpretation we provide an explanation why the Turing-Good-Robbins estimators are always biased. An iterative procedure is suggested and applied to these estimators, which leads to new estimators whose biases are reduced. Using this basic construction we are able to generalize our discussion to a much broader class of coverage problems with the species problem as a special case. Three examples are studied in detail: the species problem, the problem of estimating the volume of a convex set and the missile-coverage problem. Furthermore, we derive general (new) estimators and study their properties by applying the interpretation to the framework of the general coverage problem. It is pointed out that, as in species problem, the general estimators derived from the interpretation are usually biased, we then apply our construction together with the iterative procedure to the previous three examples to produce new estimators whose biases are reduced. Finally, we extend our construction to the conditional cases.