Nonparametric Goodness-of-fit Testing under Covariate Shift

This paper studies nonparametric goodness-of-fit testing when labeled data come from a source population but evaluation concerns a target population under covariate shift. It combines truncated importance-weighted kernel ridge regression with a multiplier bootstrap to construct confidence sets for the regression function. Truncation is designed to stabilize both estimation and bootstrap calibration, including settings with heavy-tailed target-to-source density ratios. Under operator compatibility conditions, the authors establish nonasymptotic validity and sharpness, with explicit coverage-error rates tied to density-ratio assumptions and spectral decay of the kernel operator. Numerical experiments are reported as supporting evidence.