Variance estimation and asymptotic confidence bands for the mean estimator of sampled functional data with high entropy unequal probability sampling designs
Hervé Cardot, Camelia Goga, Pauline Lardin
备注：
Revised for the Scandinavian Journal of Statistics
For fixed size sampling designs with high entropy it is well known that the variance of the Horvitz-Thompson estimator can be approximated by the Hájek formula. The interest of this asymptotic variance approximation is that it only involves the first order inclusion probabilities of the statistical units. We extend this variance formula when the variable under study is functional and we prove, under general conditions on the regularity of the individual trajectories and the sampling design, that we can get a uniformly convergent estimator of the variance function of the Horvitz-Thompson estimator of the mean function. Rates of convergence to the true variance function are given for the rejective sampling. We deduce, under conditions on the entropy of the sampling design, that it is possible to build confidence bands whose coverage is asymptotically the desired one via simulation of Gaussian processes with variance function given by the Hájek formula. Finally, the accuracy of the proposed variance estimator is evaluated on samples of electricity consumption data measured every half an hour over a period of one week.
Statistical Inference for Scenario-Based Dynamic Optimization under Uncertainty
Sandia National Laboratories（桑迪亚国家实验室）；H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院 H. 米尔顿·斯图尔特工业与系统工程学院）
Optimal Testing of Discrete Distributions with High Probability
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；MPI（马克斯·普朗克研究所）；University of California, San Diego（加利福尼亚大学圣迭戈分校）；Yale University（耶鲁大学）；UT Austin（德克萨斯大学奥斯汀分校）
