条件泊松抽样下二阶包含概率的递归计算

The Recursive Formula of the Second-order Inclusion Probabilities for the Conditional Poisson Sampling

条件泊松抽样是一种非常重要的样本容量n固定的近似rPS抽样设计,已有的许多不等概率抽样设计几乎都是泊松抽样或条件泊松抽样的发展和改进.众所周知,在不等概率抽样设计传统理论中,构造关心变量Y的总值或均值的Horvitz-Thompson估计量和该估计的方差估计,需要用到单元一阶包含概率和二阶包含概率.另外,在基于泊松抽样的各种改进设计研究中,计算包含概率时几乎都要用到条件泊松抽样的包含概率,因此有必要发展条件泊松抽样的包含概率的递推公式.本文修正了条件泊松抽样原有的一阶包含概率的递归计算公式,使其适用范围更加明确、广泛,首次给出了二阶包含概率精确的递归计算公式,并利用实例说明了所提出递推公式的可用性.得到的结果完善了条件泊松抽样理论,既具有理论意义,又具有潜在的应用价值.

The Conditional Poisson Sampling for selecting sample with fixed size n is an ap- proximate πPS sampling design without replacement. Many unequal probability sampling designs are developed or improved based on the Conditional Poisson Sampling or the Poisson Sampling. In the traditional theory of unequal probability sampling design, the knowledge of the first-order inclusion probabilities 7ri and the second-order inclusion probabilities Irij are needed to build the unbiased estimator of Horvitz-Thompson for the population total or mean of study variable Y and to obtain variance estimation of this estimator. Previous studies indicated it is necessary to use the formula of the first-order inclusion probabilities for the Conditional Poisson Sampling to calculate the inclusion probabilities for other de- signs. However, topics in developing recursive formulas of the inclusion probabilities for the Conditional Poisson Sampling have not been well investigated. This paper aims at improving recursive formulas of the first-order inclusion probabilities for the Conditional Poisson Sampling and broadening its applicable range. Remarkably, this paper presents the recursive formula of the second-order inclusion probabilities at the first time. Examples in the paper illustrate the availability of the proposed recursive formula. The obtained results significantly improve the theory of the Conditional Poisson Sampling, and exhibit potential application value.