Bootstrap Nonparametric Inference under Data Integration

We propose multiplier bootstrap procedures for nonparametric inference and uncertainty quantification of the target mean function, based on a novel framework of integrating target and source data. We begin with the relatively easier covariate shift scenario with equal target and source mean functions and propose estimation and inferential procedures through a straightforward combination of all target and source datasets. We next consider the more general and flexible distribution shift scenario with arbitrary target and source mean functions, and propose a two-step inferential procedure. First, we estimate the target-to-source differences based on separate portions of the target and source data. Second, the remaining source data are adjusted by these differences and combined with the remaining target data to perform the multiplier bootstrap procedure. Our method enables local and global inference on the target mean function without using asymptotic distributions. To justify our approach, we derive an optimal convergence rate for the nonparametric estimator and establish bootstrap consistency to estimate the asymptotic distribution of the nonparametric estimator. The proof of global bootstrap consistency involves a central limit theorem for quadratic forms with dependent variables under a conditional probability measure. Our method applies to arbitrary source and target datasets, provided that the data sizes meet a specific quantitative relationship. Simulation studies and real data analysis are provided to examine the performance of our approach.