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Variance Estimation under Two-Phase Sampling

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  • Takumi Saegusa

Abstract

type="main" xml:id="sjos12152-abs-0001"> 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.

Suggested Citation

  • Takumi Saegusa, 2015. "Variance Estimation under Two-Phase Sampling," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(4), pages 1078-1091, December.
    • Handle: RePEc:bla:scjsta:v:42:y:2015:i:4:p:1078-1091
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    File URL: http://hdl.handle.net/10.1111/sjos.12152
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    References listed on IDEAS

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    1. Thomas Lumley & Pamela A. Shaw & James Y. Dai, 2011. "Connections between Survey Calibration Estimators and Semiparametric Models for Incomplete Data," International Statistical Review, International Statistical Institute, vol. 79(2), pages 200-220, August.
    2. Norman E. Breslow & Jon A. Wellner, 2007. "Weighted Likelihood for Semiparametric Models and Two‐phase Stratified Samples, with Application to Cox Regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 34(1), pages 86-102, March.
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    1. Göran Kauermann & Mehboob Ali, 2021. "Semi-parametric regression when some (expensive) covariates are missing by design," Statistical Papers, Springer, vol. 62(4), pages 1675-1696, August.

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