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.