On Sampling Designs with Ordered Conditional Inclusion Probabilities

In this paper we consider a family of sampling designs for which increasing first-order inclusion probabilities imply, in a specific sense, increasing conditional inclusion probabilities. It is proved that the complementary Midzuno, the conditional Poisson, and the Sampford designs belong to this family. It is shown that designs of the family are more efficient than a comparable with-replacement design. Furthermore, the efficiency gain is explicitly given for these designs.