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.     

Optimal domain estimation under summation restriction

The domain estimators that do not sum up to the population total (estimated or known) are considered. In order to achieve their additivity, the theory of the general restriction (GR)-estimator [Knottnerus P., 2003. Sample Survey Theory: Some Pythagorean Perspectives. Springer, New York] is used. The elaborated domain GR-estimators are optimal, they have the minimum variance in a class of estimators that satisfy summation restriction. Furthermore, their variances are smaller than the variances of the corresponding initial domain estimators. The variance/covariance formulae of the domain GR-estimators are explicitly given.The ratio estimators as representatives of the non-additive domain estimators are considered. Their design-based covariance matrix, being crucial for the GR-estimator, is presented. Its structure simplifies under certain assumptions on sampling design (and population model). The corresponding simpler forms of the domain GR-estimators are elaborated as well. The hypergeometric [Traat I., Ilves M., 2007. The hypergeometric sampling design, theory and practice. Acta Appl. Math. 97, 311–321] and the simple random sampling designs are considered in more detail. The results are illustrated in a simulation study where the optimal domain estimator displays its superiority among other meaningful domain estimators. It is noteworthy that due to the imposed restrictions also these other estimators, though not optimal, can be much more precise than the initial estimators.