A Grid-Rate Condition for Valid Uniform Inference

TL;DR

This paper establishes a formal grid-growth condition ensuring valid uniform inference for continuous functionals, bridging the gap between heuristic rules and theoretical guarantees.

Contribution

It introduces a simple, formally justified grid-growth condition for uniform inference in nonparametric estimation within Donsker classes.

Findings

The condition L_n = ω(r_n^{1/4}) guarantees asymptotically valid inference.

The condition ensures approximation error is negligible compared to stochastic variation.

Applicable to twice continuously differentiable functions estimable at the r_n^{1/2} rate.

Abstract

Estimating a continuous functional $F: \X \to \R$ involves specifying $L_n^d$ nodes on $\X \subset \R^d$ for estimation and uniform inference. While asymptotically valid inference requires $L_n$ to increase with $n$, existing fixed-$L$ rules of thumb and heuristic data-driven approaches lack formal justification. This paper shows that, for functions within a Donsker class, the simple grid-growth condition \(L_n=\omega(r_n^{1/4})\) is sufficient for valid inference for twice continuously differentiable functions estimable at the \(r_n^{1/2}\) rate. This condition ensures that the approximation error is asymptotically negligible relative to the stochastic variation of the empirical process.