A Grid-Rate Condition for Valid Uniform Inference

2026-05-12

AI Summary

This study addresses a gap in the theoretical foundations of uniform inference for continuous functions, specifically concerning the appropriate growth rate of grid points relative to sample size. Focusing on estimators that are twice continuously differentiable and belong to a Donsker class with convergence rate \( r_n^{1/2} \), the paper establishes—for the first time—a sufficient condition requiring the number of grid points \( L_n \) to satisfy \( L_n = \omega(r_n^{1/4}) \). This condition, derived from empirical process theory and function approximation analysis, ensures that the approximation error becomes asymptotically negligible compared to stochastic fluctuations. Consequently, valid uniform inference is achieved, providing a formal theoretical guarantee that had been missing from existing heuristic approaches.

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 \(L_n=ω(r_n^{1/4})\) 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.

Problem

• uniform inference
• grid-rate condition
• Donsker class
• approximation error
• empirical process

Innovation

• grid-rate condition
• uniform inference
• Donsker class
• approximation error
• empirical process