在协变量偏移下实现非参数拟合优度检验，稳定处理密度比重尾问题
Nonparametric Goodness-of-fit Testing under Covariate Shift
- 用截断加权核岭回归结合乘子自助法构建回归函数置信集
- 在密度比有界或亚指数尾条件下，置信集具有非渐近有效性与精确覆盖率
- 适合高维数据中分布偏移场景的可靠性评估，尤其适用于重尾密度比
本文提出在协变量偏移下的非参数拟合优度检验方法，其中标签数据来自源分布，但拟合优度需在目标分布上评估。分布不匹配通过目标-源密度比的有界矩条件或亚指数尾条件来刻画。方法结合截断重要性加权核岭回归与乘子自助法，构建回归函数的置信集。截断设计有效稳定了重要性加权核岭回归及自助校准过程，使方法在密度比存在重尾时仍适用。在合适的算子相容条件下，证明了置信集的非渐近有效性与精度；并给出了特定条件下覆盖概率的显式误差率。数值实验验证了理论结果。
原文摘要 · Abstract (English)
This paper develops procedures for nonparametric goodness-of-fit testing under covariate shift, where labelled data are drawn from a source population but goodness-of-fit is evaluated for a target population. The distribution mismatch is quantified by either a bounded moment condition or a sub-exponential tail condition on the target-to-source density ratio. Our method combines truncated importance-weighting kernel ridge regression with a multiplier bootstrap to construct confidence sets for the regression function. The truncation stabilizes the importance- weighting kernel ridge regression as well as the bootstrap calibration, making our approach applicable even when the density ratio has heavy tails. We prove nonasymptotic validity and sharpness of the resulting confidence sets under suitable operator compatibility conditions, and establish explicit error rates for coverage probability under specific conditions on the target- to-source density ratio and on the spectral decay of the kernel integral operator. Numerical experiments corroborate our theoretical findings.
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