Variance Estimation under Two-Phase Sampling

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 probabili...

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Bibliographic Details
Published in:Scandinavian journal of statistics 2015-12, Vol.42 (4), p.1078-1091
Main Author: Saegusa, Takumi
Format: Article
Language:English
Subjects:
Asymptotic methods
bootstrap
Estimating techniques
sampling without replacement
semiparametric model
Statistical analysis
Studies
two-phase sampling
variance estimation
weighted likelihood
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Summary: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.
ISSN:0303-6898
1467-9469
DOI:10.1111/sjos.12152