Weighted polynomial models and weighted sampling schemes for finite population

Abstract

This paper outlines a theoretical framework for finite population models with unequal sample probabilities, along with sampling schemes for drawing random samples from these models. We first present four exact weighted sampling schemes that can be used for any finite population model to satisfy such requirements as ordered/ unordered samples, with/without replacement, and fixed/nonfixed sample size. We then introduce a new class of finite population models called weighted polynomial models or, in short, WPM. The probability density of a WPM is defined through a symmetric polynomial of the weights of the units in the sample. The WPM is shown to have been applied in many statistical analyses including survey sampling, logistic regression, case-control studies, lottery, DNA sequence alignment and MCMC simulations. We provide general strategies that can help improve the efficiency of the exact weighted sampling schemes for any given WPM. We show that under a mild condition, sampling from any WPM can be implemented within polynomial time. A Metropolis-Hasting-type scheme is proposed for approximate weighted sampling when the exact sampling schemes become intractable for moderate population and sample sizes. We show that under a mild condition, the average acceptance rate of the approximate sampling scheme for any WPM can be expressed in closed form using only the inclusion probabilities.