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Variance Estimation under Two‐Phase Sampling
Author(s) -
Saegusa Takumi
Publication year - 2015
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/sjos.12152
Subject(s) - mathematics , estimator , statistics , variance (accounting) , delta method , variance function , consistency (knowledge bases) , sampling (signal processing) , sample variance , population variance , population , stratified sampling , sample (material) , variance based sensitivity analysis , one way analysis of variance , analysis of variance , computer science , accounting , filter (signal processing) , business , computer vision , chemistry , geometry , demography , chromatography , sociology
We consider the variance estimation of the weighted likelihood estimator (WLE) under two‐phase stratified sampling without replacement. Asymptotic variance of the WLE in many semiparametric models contains unknown functions or does not have a closed form. The standard method of the inverse probability weighted (IPW) sample variances of an estimated influence function is then not available in these models. To address this issue, we develop the variance estimation procedure for the WLE in a general semiparametric model. The phase I variance is estimated by taking a numerical derivative of the IPW log likelihood. The phase II variance is estimated based on the bootstrap for a stratified sample in a finite population. Despite a theoretical difficulty of dependent observations due to sampling without replacement, we establish the (bootstrap) consistency of our estimators. Finite sample properties of our method are illustrated in a simulation study.