  Instrumental variable methods allow for inference about the treatment effect
by controlling for unmeasured confounding in randomized experiments with
noncompliance. However, many studies do not consider the observed compliance
behavior in the testing procedure, which can lead to a loss of power. In this
paper, we propose a novel nonparametric likelihood approach, referred to as the
binomial likelihood (BL) method, that incorporates information on compliance
behavior while overcoming several limitations of previous techniques and
utilizing the advantages of likelihood methods. Our proposed method produces
proper estimates of the counterfactual distribution functions by maximizing the
binomial likelihood over the space of distribution functions. Using this we
propose two versions of a binomial likelihood ratio test for the null
hypothesis of no treatment effect. We show that both versions are more powerful
to detect any distributional change than existing methods in finite sample
cases, and are asymptotically equivalent to the two-sample Anderson-Darling
test. We also develop an efficient algorithm for computing our estimates, and
apply the binomial likelihood method to a study of the effect of Medicaid
coverage on mental health using the Oregon Health Insurance Experiment.
