Case-Control Studies with Complex Sampling:
Asymptotics, Sampling Proportional to Size, and
Local Central Limit Theorems


Larry Goldstein, Department of Mathematics, USC
Joint with Richard Arratia, Department of Mathematics, USC
and Bryan Langholz, Department of Preventive Medicine, USC



In epidemiological studies, sampling designs such as counter matching were
originally developed in semi-parametric settings where a cohort of
individuals with a common base line hazard function
is followed over time in order to detect relationships between disease and
exposure. Such designs can also be implemented in studies having no time
dependence, but martingale methods and counting process techniques can no
longer be used in the study of estimator properties. In their place, local
central limit theorems for independent but not identically distributed
Bernoulli random variables must be developed, which by
conditioning, allows for the study of asymptotic estimator properties and
also gives results about the high correlation structure
in various sampling proportional to size type schemes.