On estimating the conditional probability of discovering a new species

Based on a capture-recapture sample of size $i;n+k,n≥ l,k ≥ 0, from a population of an unknown number of distinct species (or classes), the problem of estimating the total probability of the species unobserved in the first n selections is considered. As the estimand depends on both the unknown parameters and the data, the standard theory of estimation is inadequate for this problem A suitable definition of sufficiency is introduced and used to prove a Rao-Blackwell type result and discuss uniformly minimum mean squared error unbiased estimation. An alternative proof for an inadmissibility result is presented. The new proof gives more insight and a method for deriving improved estimators. The theoretical developments may be useful in other problems concerning inferences about random parametric functions.