Complex sampling designs: Uniform limit theorems and applications

In this paper, we develop a general approach to proving global and local uniform limit theorems for the Horvitz–Thompson empirical process arising from complex sampling designs. Global such as Glivenko–Cantelli Donsker theorems, asymptotic modulus related ratio-type are proved both process, its calibrated version. Limit of other variants their conditional versions also established. Our reveals an interesting feature: problem deriving is essentially no harder than establishing corresponding finite-dimensional once usual complexity conditions on function class satisfied. These then applied important statistical problems including (i) $M$-estimation, (ii) $Z$-estimation (iii) frequentist theory pseudo-Bayes procedures, all with weighted likelihood, illustrate wide applicability.