Nonparametric Goodness-of-fit Testing under Covariate Shift

2026-08-05 · 13:51 UTC ·Zhen Hou, Dong Xia ·1 min read

Brief

Nonparametric goodness-of-fit testing under covariate shift: the authors handle label/source mismatch by truncating importance weights in kernel ridge regression and using a multiplier bootstrap to form confidence sets for the regression function. They establish nonasymptotic validity and sharpness under operator-compatibility assumptions, give explicit coverage error rates tied to density-ratio tails and kernel spectral decay, and validate results numerically. Full text is available from the arXiv link.

Why it matters

Hou & Xia (2026, arXiv:2608.04860v1) propose a testing procedure for nonparametric goodness-of-fit under covariate shift that builds confidence sets for the regression function using truncated importance-weighted kernel ridge regression combined with a multiplier bootstrap; truncation stabilizes both estimator and bootstrap under heavy-tailed density ratios.

Key details

They prove nonasymptotic validity and sharpness of the confidence sets under operator-compatibility conditions and derive explicit error rates for coverage probability depending on the target-to-source density-ratio assumptions (bounded moments or sub-exponential tails) and the spectral decay of the kernel integral operator; numerical experiments support the theory.

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.