Estimation in surveys using conditional inclusion probabilities: Complex design

This paper investigates a repeated sampling approach to take into account auxiliary information in order to improve the precision of estimators. The objective is to build an estimator with a small conditional bias by weighting the observed values by the inverses of the conditional inclusion probabilities. A general approximation is proposed in cases when the auxiliary static is a vector of Horvitz-Thompson estimators. This approximation is quite close to the optimal estimator discussed by Fuller and Isaki (1981), Montanari (1987, 1997), Deville (1992) and Rao (1994, 1997). Next, the optimal estimator is applied to a stratified sampling design and it is shown that the optimal estimator can be viewed as a generalised regression estimator for which the stratification indicator variables are also used at the estimation stage. Finally, the application field of this estimator is discussed in the general context of the use of auxiliary information.