Advances in Model-agnostic Approaches to Statistical Inference

Abstract

This dissertation focuses broadly on contributing to understanding the impact of incorrect modeling assumptions on analyses and arguing for the use of methods for statistical inference that are valid under weaker conditions than required for traditional approaches. By statistical inference, we mean providing both a point estimate for the population value of a parameter as well as valid confidence intervals and hypothesis tests. In settings involving coarsened data or nuisance parameters that are difficult to estimate, model-based approaches to regression can provide misleading results when the model fails to hold. Through the examples of the partially linear additive model and Cox proportional hazards model, we provide guidance for evaluating the properties of model-based regressions and illustrate alternative model-agnostic approaches that avoid undesirable behaviors. We introduce a novel expansion of a remainder term to derive a framework to obtain doubly robust inference for a broad class of parameters. This work extends recent nonparametric methods to achieve doubly robust inference -- rather than simply doubly robust estimation -- for the average treatment effect specifically. While estimation of quantiles is not much more difficult than for means, construction of confidence intervals presents greater challenges. Hence, we study and evaluate several model-agnostic procedures to obtain confidence regions and hypothesis tests for quantiles.