Horvitz-Thompson estimators for functional data: asymptotic confidence bands and optimal allocation for stratified sampling

• 成果类型: Article
• 署名作者: Cardot, Herve; Josserand, Etienne
• 署名单位: Universite Bourgogne Europe; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asq070
• 发表日期: 2011
• 页码: 107118
• 关键词: normality; DESIGN

摘要

When dealing with very large datasets of functional data, survey sampling approaches are useful in order to obtain estimators of simple functional quantities, without being obliged to store all the data. We propose a Horvitz-Thompson estimator of the mean trajectory. In the context of a superpopulation framework, we prove, under mild regularity conditions, that we obtain uniformly consistent estimators of the mean function and of its variance function. With additional assumptions on the sampling design we state a functional central limit theorem and obtain asymptotic confidence bands. Stratified sampling is studied in detail, and we also obtain a functional version of the usual optimal allocation rule, considering a mean variance criterion. These techniques are illustrated by a test population of N=18 902 electricity meters for which we have individual electricity consumption measures every 30 minutes over one week. We show that stratification can substantially improve both the accuracy of the estimators and reduce the width of the global confidence bands compared with simple random sampling without replacement.