  Under "measurement constraints," responses are expensive to measure and
initially unavailable on most of records in the dataset, but the covariates are
available for the entire dataset. Our goal is to sample a relatively small
portion of the dataset where the expensive responses will be measured and the
resultant sampling estimator is statistically efficient. Measurement
constraints require the sampling probabilities can only depend on a very small
set of the responses. A sampling procedure that uses responses at most only on
a small pilot sample will be called "response-free." We propose a response-free
sampling procedure \mbox{(OSUMC)} for generalized linear models (GLMs). Using
the A-optimality criterion, i.e., the trace of the asymptotic variance, the
resultant estimator is statistically efficient within a class of sampling
estimators. We establish the unconditional asymptotic distribution of a general
class of response-free sampling estimators. This result is novel compared with
the existing conditional results obtained by conditioning on both covariates
and responses. Under our unconditional framework, the subsamples are no longer
independent and new martingale techniques are developed for our asymptotic
theory. We further derive the A-optimal response-free sampling distribution.
Since this distribution depends on population level quantities, we propose the
Optimal Sampling Under Measurement Constraints (OSUMC) algorithm to approximate
the theoretical optimal sampling. Finally, we conduct an intensive empirical
study to demonstrate the advantages of OSUMC algorithm over existing methods in
both statistical and computational perspectives.
