Optimal Weights for double Many-To-One Generalized Weight Share Method

In probabilistic sampling, a sampling frame comprising the target population may not be available. However, indirect sampling can be used when the units on a sampling frame can be linked to the target population, with the values observed in the target population used for estimation. If the target population can only be linked to the sampling frame through an intermediate population, then double indirect sampling is a viable option. In this case, the sampling weights can be obtained with the double generalized weight share method (GWSM). The double GWSM reduces the number of observed links compared to the simple GWSM but can detrimentally affect the precision of the estimator, as in the example of the French postal service (La Poste). In this article, we show the existence of a set of sampling weights that minimize the variance of the double GWSM in specific cases with La Poste as exemplar. These weights ensure that the implementation advantages of the double GWSM are not offset by a loss of precision. When these weights cannot be computed, as is the case in the French postal traffic survey, we propose alternative weights that can be used to improve the precision of the double GWSM estimator. Results are illustrated through Monte Carlo simulations and an application to the French postal traffic survey.