Weighted likelihood estimation under two-phase sampling

Details Weighted likelihood estimation under two-phase sampling Weighted likelihood estimation under two-phase sampling

2011-12-21

arxiv.org/abs/1112.4951v2

Mathematics

Statistics Theory

57 pages

Scientific paper

We develop asymptotic theory for weighted likelihood estimators (WLE) under two-phase stratified sampling without replacement. We also consider several variants of WLE's 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 $Z$-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 non-regular 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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