A ‘Robust’ Framework for Statistical Inference

ARUN KUMAR РCARNEGIE 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).’

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