In the era of digital information, people are faced with data that may not only be large-scale but also heterogeneous. In this paper, we study statistical inference for the overall population mean function of large-scale multi-source heterogeneous datasets. By borrowing hierarchical sampling methods and divide-and-conquer techniques, we propose a weighted local linear estimator for the overall population mean function of multi-source heterogeneous data. Through studying the pointwise convergence properties and extreme value distribution properties of the estimator, we construct asymptotically accurate simultaneous confidence bands and pointwise confidence intervals for large-scale multi-source heterogeneous data. Our proposed methods are applicable not only to scenarios of heterogeneous data but also to scenarios of homogeneous data using divide-and-conquer methods. Numerical simulation studies show that the proposed methods perform well in analyzing both large-scale multi-source heterogeneous data and homogeneous data. As an illustration, we apply the proposed methods to hypothesis testing problems on Beijing multi-site air-quality data and U.S. census data.

In this paper, we consider a weighted local linear estimator based on the inverse selection probability for nonparametric regression with missing covariates at random. The asymptotic distribution of the maximal deviation between the estimator and the true regression function is derived and an asymptotically accurate simultaneous confidence band is constructed. The estimator for the regression function is shown to be oracally efficient in the sense that it is uniformly indistinguishable from that when the selection probabilities are known. Finite sample performance is examined via simulation studies which support our asymptotic theory. The proposed method is demonstrated via an analysis of a data set from the Canada 2010/2011 Youth Student Survey.