Nonparametric and Semiparametric Estimation of Instrumental Variable Method

Abstract

The instrumental variable approach has been widely used for estimating the treatment effect in the presence of unmeasured confounding, e.g. randomized trials with noncompliance problems and observational studies. While most literature focus on the estimation of compliers averaged causal effect (CACE) nonparametrically or based on parametric assumptions, under the IV assumptions, fewer works focus on estimating distributional causal effect using IV. We study a novel monotone cumulative distribution function estimator of an outcome variable for compliers receiving treatment or control. The estimation procedures involve a weighted quantile regression and a post-estimation rearrangement adjustment. We show that the proposed estimator is consistent and develop large sample properties. Based on the asymptotic properties of the proposed estimator, a Wilcoxon-type statistic is proposed to test the equivalence of CDF for compliers receiving treatment and control. By comparing the influence function of the proposed estimator to the efficient influence function, we modify the proposed estimator and obtain a local efficient and robust estimator in the sense that when the unknown density functions are correctly specified, it reaches the semiparametric efficiency bound and when the unknown density functions are misspecified, it is still a consistent estimator. For the censoring outcomes, we propose a method to estimate quantile functions and survival functions for potential outcomes under independent censoring and noncompliance. Based on the martingale feature associated with the censoring data, we estimate quantile functions for compliers. Then using the possibly non-monotone quantile function, we construct a monotone and bounded estimator for the survival function. By using empirical process techniques, we establish asymptotic properties, including uniform consistency and weak convergence for the proposed estimators. For general observational studies with unmeasured confounding problems, we impose a no-interaction assumption proposed by Wang and Tchetgen Tchetgen (2018) and propose a new class of IV models that identify quantities of potential outcomes for the whole population. Our work complements current research on using instrumental variable method to estimate distributions of potential outcomes and infer heterogenous treatment effect for observational studies in the presence of unmeasured confounding, especially for the censoring outcomes. Simulation results, real data examples, and proofs are detailed in this dissertation.