Semiparametric Inference with Incomplete Data: Data Fusion, Instrumental Variables, and Proximal Causal Models

Abstract

This dissertation develops semiparametric theory for two settings linked by their reliance on incomplete data. In Chapter 1, we take steps towards a unified theory of semiparametric inference for fused data, where practitioners observe independent samples from several distinct sources, each informative about a parameter of interest in a target population. We give unified methods for computing the (efficient) influence functions for smooth parameters when each source's observed-data distribution is assumed to align with certain marginal and conditional distributions of a source-specific factorization of the joint target distribution, paving the way for machine-learning debiased, semiparametric efficient estimation. In Chapter 2, we turn to parameters defined as continuous linear functionals of solutions to ill-posed inverse problems, a class including non-parametric instrumental variables, proximal causal inference, and shadow variables. Under the minimal assumptions permitting pointwise asymptotically valid debiased inference based on Neyman-orthogonal estimating equations, we show these parameters are highly discontinuous in the data generating process. Consequently, locally uniformly valid inference is impossible: no locally uniformly consistent estimator exists, and any locally uniformly honest confidence interval has diameter diverging with the sample size. We further derive conditions restoring continuity, and show that these parameters may still fail to be pathwise differentiable, rendering all asymptotically linear estimators irregular and beyond the reach of semiparametric efficiency theory.