COMPLEX SAMPLING DESIGNS: UNIFORM LIMIT THEOREMS AND APPLICATIONS

• 成果类型: Article
• 署名作者: Han, Qiyang; Wellner, Jon A.
• 署名单位: Rutgers University System; Rutgers University New Brunswick; University of Washington; University of Washington Seattle
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/20-AOS1964
• 发表日期: 2021
• 页码: 459-485
• 关键词: 2-phase stratified samples; von mises theorems; Empirical Processes; Semiparametric models; weighted likelihood; convergence-rates; ASYMPTOTIC THEORY; calibration estimators; varying probabilities; regression-estimators

摘要

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