The Annals of Statistics

Nonparametric Inference Under Biased Sampling from a Finite Population

Peter J. Bickel, Vijayan N. Nair, and Paul C. C. Wang

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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.

Article information

Ann. Statist., Volume 20, Number 2 (1992), 853-878.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62D05: Sampling theory, sample surveys 62G20: Asymptotic properties 62P99: None of the above, but in this section

Asymptotically efficient informative sampling maximum likelihood petroleum resource estimation probability proportional to size size-biased sampling


Bickel, Peter J.; Nair, Vijayan N.; Wang, Paul C. C. Nonparametric Inference Under Biased Sampling from a Finite Population. Ann. Statist. 20 (1992), no. 2, 853--878. doi:10.1214/aos/1176348660.

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