In a recent paper, Boistard, Lopuhaä and Ruiz-Gazen proved some functional central limit theorems for single-stage sampling designs. An application of the results, in combination with the functional delta method, is the obtention of the limiting distribution of estimators of some complex parameters such as the poverty rate. These functional central limit theorems are derived for different types of empirical processes obtained from suitably centering the Horvitz-Thompson and HaÌjeÌ€k empirical distribution functions. These results are obtained merely under conditions on higher order inclusion probabilities corresponding to the sampling design at hand. This makes the results generally applicable and allows more complex sampling designs that go beyond the classical simple random sampling without replacement or Poisson sampling.

It is possible to generalize the results to single-stage stratified sampling designs. We ï¬rst consider the situation where the number of strata is ï¬xed. In this case, we can represent the Horvitz-Thompson empirical process for the whole population as a ï¬nite sum of weighted Horvitz-Thompson empirical processes for the different strata, and similarly for the HaÌjeÌ€k empirical process. In order to use the results from the paper by Boistard, Lopuhaä and Ruiz-Gazen, we impose the assumptions, required for a functional central limit theorem, on each individual stratum. This establishes a functional central limit theorem for the individual empirical processes on each stratum. With some minor additional assumptions on the weights this leads to a functional central limit theorem for the overall empirical processes. The above approach is direct and short, making use of the independence between the strata and the fact that the number of strata is ï¬xed. But in order to allow the number of strata to depend on the size of the population, including the case where the number of strata tends to inï¬nity, we need to use a different approach. We propose to impose assumptions on each stratum, as weak as possible, but such that they imply the conditions needed in the paper by Boistard, Lopuhaä and Ruiz-Gazen for the overall empirical process.

This work is a joint work with H.P. Lopuhaä