Nonparametric Estimation of the Probability of Discovering a New Species

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Abstract
A random sample is taken from a population consisting of an unknown number of distinct species. A quantity of interest is the probability of discovering a new species when an additional draw from the population is made. An estimator of this quantity was introduced by Starr (1979). We prove a conjecture of Starr's that the estimator is uniformly minimum variance unbiased and give various asymptotic properties of the estimator. A nonparametric maximum likelihood estimator that has similar asymptotic properties is introduced. A Monte Carlo study that suggests guidelines for choosing an estimator under various circumstances is given. To amplify, suppose that if we take a sample of size 1 from a population then the probability of drawing a representative of the ith species is p 1. If n draws are made with replacement, the (unconditional) probability that species i will be observed for the first time on the n + 1st draw is given simply as the term p i q n i (q i = 1 − p i ) from a geometric distribution. Consequently, θ n = Σ i p i q n i represents the (unconditional) probability that some new species will be drawn for the first time on the n + 1st draw. No unbiased estimate of θ n , can be derived from a sample of size n, but if one additional observation is allowed then such an estimator arises naturally. Denoted by V 1, this estimator is simply the number of species observed only once divided by n + 1, the number of draws. This estimator was proposed by Robbins (1968). An analogous unbiased estimator, Vm , was proposed by Starr (1979) for the case in which m additional draws are made. Starr conjectured that Vm , is a minimum variance unbiased estimate. We prove Starr's conjecture, using the theory of U statistics. This theory can also be used to show that V m is asymptotically normally distributed as m → ∞ and that the rate of convergence is faster if the p i 's are all equal. As an alternative to estimating θ i by V 1, we consider estimating p i , by the fraction of times species i is observed in n + 1 draws, say , and then estimating θn, by . We also construct a similar estimator when m additional draws are made. Although this nonparametric maximum likelihood estimator is biased for small samples, we show that it has asymptotic properties similar to Vm : it is asymptotically unbiased and has the same asymptotic distribution. Moreover, by way of simulations and special cases, we show that can dominate Vm in terms of mean squared error.

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