  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.
