Orthogonal Statistical Learning and Inference for Function-Valued Parameters with Heterogeneous Data Sources

Abstract

This dissertation develops methods for estimation of and inference on function-valued parameters when the available data are heterogeneous, drawn from partially aligned sources or missing at random by study design. Neyman-orthogonal losses and one-step estimation allow nuisance functions to be estimated with flexible machine learning methods without compromising the first-order behavior of the resulting estimators and statistical tests. Chapter 1 studies estimation of causal dose-response functions by combining partially aligned data sources. I propose a data fusion framework that estimates the population risk of the dose-response functions using a Neyman-orthogonal loss. Then I show that data fusion necessarily improves worst-case performance in a minimax sense. Chapter 2 develops nonparametric tests of whether a function-valued parameter belongs to a closed linear subspace, such as the space of linear functions or of additive functions. Each null hypothesis is membership in a subspace, and I construct asymptotically linear estimators of the kernel embedding of the projected parameter. Chapter 3 provides a general orthogonal statistical learning framework for function-valued parameters under two-phase sampling. The argument of the function is observed only on a gold-standard subsample. I propose a corrected loss that preserves Neyman orthogonality, and the price of the incomplete design is shown to be second order.