## Weighted likelihood estimation under two-phase sampling.(English)Zbl 1347.62033

Summary: 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.

### MSC:

 62E20 Asymptotic distribution theory in statistics 62G20 Asymptotic properties of nonparametric inference 62D05 Sampling theory, sample surveys 62N01 Censored data models

Survey
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### References:

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