Weighted likelihood estimation under two-phase sampling

TL;DR

This paper develops asymptotic theory for weighted likelihood estimators in two-phase stratified sampling, including variants with estimated weights and calibration, with applications to survival analysis models.

Contribution

It introduces new empirical process tools and derives asymptotic distributions for WLEs in complex sampling schemes, extending semiparametric inference methods.

Findings

Asymptotic distributions of WLEs are derived for various sampling schemes.

Comparison of methods shows differences in asymptotic variances.

Theoretical results are illustrated with Cox models under censoring.

Abstract

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.