A ‘Robust’ Framework for Statistical Inference

ARUN KUMAR – CARNEGIE MELLON UNIVERSITY 

ABSTRACT

Confidence intervals (and hypothesis tests) are fundamental components of statistical analysis, integral to any rigorous scientific study. The traditional framework for constructing confidence intervals for a functional/parameter $\theta_0$ starts with an estimator, denoted as $\widehat{\theta}_n$, possessing a known rate of convergence and an estimable limiting distribution. Resampling techniques, such as bootstrap and subsampling, have been introduced to relax the assumption of a known convergence rate and to provide estimates of limiting distributions.

However, there are still scenarios that elude analysis through resampling techniques. In this presentation, I propose a robust framework for statistical inference, with ‘robust’ being interpreted as resilient to distributional assumptions. The recently introduced HulC methodology can be viewed as a special case within this framework. Despite a slight loss in efficiency, this proposed framework can offer elegant solutions to a variety of complex inference problems, including confidence intervals for online algorithms, cube-root estimators, shape-constrained estimators, non-/semi-parametric estimators, and non-standard regression problems.

The foundation of this talk rests on concepts developed in my recent works, namely, ‘The HulC: Confidence Regions from Convex Hulls (2023+, JRSS-B)’ and ‘Median Regularity and Honest Inference (2023, Biometrika).’

Related Papers: 

• https://arxiv.org/abs/2311.16598
• https://arxiv.org/abs/2105.14577
• https://arxiv.org/abs/2206.02954