Robust Inference: A Price of Misspecification and How to be Resilient

 In statistical inference, it is rarely realistic that the hypothesized statistical model is well-specified to contain the target of inference. This dissertation aims to identify the statistical price of such misspecification on inferential procedures and to seek robust remedies in two focal points: Universal Inference, and Functional Estimation.

When the model is misspecified, the natural target of inference is a projection of the data-generating distribution onto the model. The first part of the dissertation presents a general method for constructing a uniformly valid confidence set for the projection under weak assumptions inspired by the universal inference approach. We provide concrete settings in which our methods yield either exact or approximate confidence sets for various projection distributions. We also investigate the rates at which these confidence sets shrink around the target of inference.

Another theme of this dissertation is to quantify the statistical price of estimating an integral functional of a density in the presence of contamination. We derive the minimax-optimal rate of estimating quadratic functionals and study to what extent the known structure of the contamination influences the risk. We discuss a general debiasing approach that can be applied when the estimator, based on high-order influence functions, achieves minimax optimality without contamination. In addition, we study adaptive rates to the unknown proportion of contamination and smoothness.