ALGORITHMS TO FIND EXACT INCLUSION PROBABILITIES FOR 2Pπps SAMPLING DESIGNS

The statistical literature contains several proposals for methods generating fixed-size without-replacement pi ps sampling designs. pi ps designs with fixed size have rarely been used due to difficulties with implementation. Recently, a new method was proposed viz. the 2P pi ps design using a two-phase approach. It was shown that the first-order inclusion probabilities of the 2P pi ps design are asymptotically equal to the target inclusion probabilities of a p pi s design. This paper extends the work on the 2P pi ps design and presents algorithms for calculation of exact first-and second-order inclusion probabilities. Starting from a probability mass function (pmf) of the sum of N independent, but not equally distributed Bernoulli variables, the algorithms are based on derived expressions for the pmfs of sums of N - 1 and N - 2 variables, respectively. Exact inclusion probabilities facilitate standard-based inference and provide a tool for studying the properties of the 2P pi ps design. Furthermore, empirical results presented show that the properties of the suggested point estimator can be improved using a more general 2P pi ps design. In addition, the frequently used Conditional Poisson sampling design is shown to be a special case of this more general 2P pi ps design.