Empirical processes in survey sampling

Patrice Bertail (1) , Emilie Chautru (2) , Stéphan Clémençon (3)

1 MODAL'X - Modélisation aléatoire de Paris X
2 AGM - UMR 8088 - Analyse, Géométrie et Modélisation
3 LTCI - Laboratoire Traitement et Communication de l'Information

It is the main purpose of this paper to study the asymptotics of variants of the empirical process in the context of survey data. Precisely, a functional central limit theorem is established when the sample is picked by means of a Poisson survey scheme. This preliminary result is then extended to the case of the rejective or conditional Poisson sampling case, and in turn to high entropy survey sampling plans which are close to the rejective design in the sense of the Bounded-Lipschitz distance. The framework we develop encompasses survey sampling designs with non-uniform first order inclusion probabilities, which can be defi ned so as to optimize estimation accuracy. Applications to Hadamard- and Fr échet-diff erentiable functionals are also considered together with the construction of uniform confi dence bands of the cumulative distribution function. Related simulation results are displayed for illustration purpose.

Mots clés: empirical processes; survey sampling; rejective scheme; Poisson scheme; functional central limit theorem; coupling

Domaines: Théorie [stat.TH]

Patrice Bertail, Emilie Chautru, Stéphan Clémençon. Empirical processes in survey sampling. 2013. ⟨hal-00989585⟩