  Weighted likelihood, in which one solves Horvitz-Thompson or inverse
probability weighted (IPW) versions of the likelihood equations, offers a
simple and robust method for fitting models to two phase stratified samples. We
consider semiparametric models for which solution of infinite dimensional
estimating equations leads to $\sqrt{N}$ consistent and asymptotically Gaussian
estimators of both Euclidean and nonparametric parameters. If the phase two
sample is selected via Bernoulli (i.i.d.) sampling with known sampling
probabilities, standard estimating equation theory shows that the influence
function for the weighted likelihood estimator of the Euclidean parameter is
the IPW version of the ordinary influence function. By proving weak convergence
of the IPW empirical process, and borrowing results on weighted bootstrap
empirical processes, we derive a parallel asymptotic expansion for finite
population stratified sampling. Whereas the asymptotic variance for Bernoulli
sampling involves the within strata second moments of the influence function,
for finite population stratified sampling it involves only the within strata
variances. The latter asymptotic variance also arises when the observed
sampling fractions are used as estimates of those known a priori. A general
procedure is proposed for fitting semiparametric models with estimated weights
to two phase data. Several of our key results have already been derived for the
special case of Cox regression with stratified case-cohort studies, other
complex survey designs and missing data problems more generally. This paper is
intended to help place this previous work in appropriate context and to pave
the way for applications to other models.
