Assumption-Lean Honest Inference for $Z$-functionals

TL;DR

This paper introduces a new assumption-lean framework for constructing honest, uniformly valid confidence sets for $Z$-functionals, using self-normalized statistics and test inversion, applicable in high-dimensional regression contexts.

Contribution

It develops a novel, assumption-lean method combining self-normalized statistics with test inversion for valid inference without explicit variance estimation.

Findings

Achieves valid coverage in high-dimensional regression.

Provides non-asymptotic bounds on confidence set width.

Outperforms classical methods like Wald and bootstrap in simulations.

Abstract

We develop a general assumption-lean framework for constructing uniformly valid confidence sets for functionals defined by moment equalities, referred to as $Z$-functionals. Our approach combines self-normalized statistics with a test inversion principle, enabling honest inference under mild regularity conditions and without explicit variance estimation. To enhance geometric tractability, we propose novel split-normalized and Gateaux-normalized statistics that yield computationally feasible and interpretable confidence sets. A central contribution of this work is a comprehensive non-asymptotic width analysis: we derive high-probability upper bounds on the diameter of the proposed confidence sets, and quantify their proximity to Wald intervals under minimal assumptions. Applications to high-dimensional non-sparse linear and generalized linear regression demonstrate that our procedures achieve valid coverage and near-optimal rate of convergence for the width/diameter, while the classical methods including Wald and bootstrap fail.