Functional central limit theorems for single-stage sampling designs

H. Boistard, H.P. Lopuhaä and A. Ruiz-Gazen,

to appear in the Annals of Statistics.

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For a joint model-based and design-based inference, we establish functional central limit theorems for the Horvitz-Thompson empirical process and the Hájek empirical process centered by their finite population mean as well as by their super-population mean in a survey sampling framework. The results apply to generic sampling designs and essentially only require conditions on higher order correlations. We apply our main results to a Hadamard differentiable statistical functional and illustrate its limit behavior by means of a computer simulation.