Nonparametric Inference Under Biased Sampling from a Finite Population

Abstract

We consider a biased sampling model that has been found useful in incorporating size biases inherent in many types of discovery data. The model postulates that the data are obtained from a finite population by selecting successively without replacement and with probability proportional to some measure of size. Unlike the ppswor scheme in survey sampling, it is assumed here that the size measure is a function of the unknown population values. In this article, we consider maximum likelihood estimation of the finite population parameters under this biased sampling model. We study the large sample behavior of the MLE's and derive a simple, asymptotically efficient approximation to the MLE. The approximate MLE is structurally similar to the Horvitz-Thompson estimator. We show that information about the order in the sample can be used to make inference even when the population size is unknown, which in fact can be estimated. Small sample behavior of the estimators is investigated through a limited simulation study, and the results are used to analyze oil and gas discovery data from the North Sea basin.