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Variance estimation for the infinite-order regime within the proposed framework

Develop consistent, implementable variance estimators for incomplete U-statistics in the infinite-order regime (k→∞) tailored to deterministic equireplicate designs, enabling practical application of the central limit theorem in Theorem \ref{theo:CLT}.

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Background

Applying the central limit theorem in the infinite-order regime requires consistent variance estimation when the kernel order k diverges. While prior works propose variance estimators for incomplete U-statistics with diverging order, integration with the paper’s equireplicate-design framework remains unaddressed.

Providing variance estimators compatible with deterministic equireplicate constructions would allow practical use of the IOUS asymptotics developed here and broaden applicability to ensemble learning and other high-order settings.

References

To apply Theorem~\ref{theo:CLT} in the infinite-order regime, we need a consistent estimator of the variance when $k$ diverges. Existing works have proposed variance estimation methods for incomplete U-statistics in the infinite-order regime, which can be applied \citep{wang2014variance,SongChenKato2019,xu_var_est_rf_2024}. However, specific details for implementation of our framework in the infinite-order regime is left for future work.

— Incomplete U-Statistics of Equireplicate Designs: Berry-Esseen Bound and Efficient Construction (2510.20755 - Miglioli et al., 23 Oct 2025) in Section 3 (Normal Approximations), Remark “Variance estimation in the infinite-order regime”