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February 2013 Weighted likelihood estimation under two-phase sampling
Takumi Saegusa, Jon A. Wellner
Ann. Statist. 41(1): 269-295 (February 2013). DOI: 10.1214/12-AOS1073


We develop asymptotic theory for weighted likelihood estimators (WLE) under two-phase stratified sampling without replacement. We also consider several variants of WLEs involving estimated weights and calibration. A set of empirical process tools are developed including a Glivenko–Cantelli theorem, a theorem for rates of convergence of $M$-estimators, and a Donsker theorem for the inverse probability weighted empirical processes under two-phase sampling and sampling without replacement at the second phase. Using these general results, we derive asymptotic distributions of the WLE of a finite-dimensional parameter in a general semiparametric model where an estimator of a nuisance parameter is estimable either at regular or nonregular rates. We illustrate these results and methods in the Cox model with right censoring and interval censoring. We compare the methods via their asymptotic variances under both sampling without replacement and the more usual (and easier to analyze) assumption of Bernoulli sampling at the second phase.


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Takumi Saegusa. Jon A. Wellner. "Weighted likelihood estimation under two-phase sampling." Ann. Statist. 41 (1) 269 - 295, February 2013.


Published: February 2013
First available in Project Euclid: 26 March 2013

zbMATH: 1347.62033
MathSciNet: MR3059418
Digital Object Identifier: 10.1214/12-AOS1073

Primary: 62E20
Secondary: 62D99 , 62G20 , 62N01

Keywords: Calibration , estimated weights , nonregular , regular , Semiparametric model , weighted likelihood

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 1 • February 2013
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