Weighted polynomial models and weighted sampling schemes for finite population
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有限总体的加权多项式模型与加权抽样方案
- 研究问题
- 对于有限总体中具有不等抽取概率的模型，如何统一构造精确与近似加权抽样，并刻画这些方案的计算可行性？
- 方法与思路
- 作者提出四种精确抽样方案，以样本单位权重的对称多项式定义加权多项式模型，并构造Metropolis–Hastings型近似方案。
- 材料陈述的贡献
- 摘要称，在一定条件下此类模型可于多项式时间内抽样，近似方案的平均接受率可仅用入样概率作闭式表达。
- 阅读关注点
- 应核对模型概率的定义、四种方案对应的抽样要求、复杂度结论的条件，以及近似方案的接受率和实际效率。
- 阅读边界
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原文摘要
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.