Abstract
We study the problem of distributional approximations to high-dimensional non-degenerate $U$-statistics with random kernels of diverging orders. Infinite-order $U$-statistics (IOUS) are a useful tool for constructing simultaneous prediction intervals that quantify the uncertainty of ensemble methods such as subbagging and random forests. A major obstacle in using the IOUS is their computational intractability when the sample size and/or order are large. In this article, we derive non-asymptotic Gaussian approximation error bounds for an incomplete version of the IOUS with a random kernel. We also study data-driven inferential methods for the incomplete IOUS via bootstraps and develop their statistical and computational guarantees.
Citation
Yanglei Song. Xiaohui Chen. Kengo Kato. "Approximating high-dimensional infinite-order $U$-statistics: Statistical and computational guarantees." Electron. J. Statist. 13 (2) 4794 - 4848, 2019. https://doi.org/10.1214/19-EJS1643
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