Nonparametric Goodness-of-fit Testing under Covariate Shift

2026-08-05

AI Summary

This work addresses the problem of nonparametric goodness-of-fit testing for the target distribution under covariate shift, where labels are available only from the source distribution. The authors propose a novel approach based on truncated importance-weighted kernel ridge regression combined with a multiplier bootstrap procedure. By introducing a truncation mechanism to stabilize importance weights under heavy-tailed density ratios and employing bootstrap calibration to construct confidence sets for the regression function, the method achieves both theoretical rigor and practical efficacy. Under an operator compatibility condition, explicit non-asymptotic coverage error bounds are established when the density ratio satisfies either bounded moment or sub-exponential tail assumptions. Numerical experiments demonstrate the superior performance of the proposed method.

Abstract

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.

Problem

• nonparametric goodness-of-fit testing
• covariate shift
• distribution mismatch
• importance weighting
• confidence sets

Innovation

• covariate shift
• nonparametric goodness-of-fit
• importance weighting
• kernel ridge regression
• multiplier bootstrap