Rejoinder on “A selective review of statistical methods using calibration information from similar studies”

We thank Professor Jun Shao for organizing this interesting discussion. We also thank the six discussants formany insightful comments and suggestions. Assembling data from different sources has been becoming a very popular topic nowadays. In our review paper, we have mainly discussed many integration methods when internal data and external data share a common distribution, though the external data may not have information for some underlying variables collected in the internal study. Indeed the common distribution assumption is very strong in practical applications. Due to the technology advance, the collection of data is gettingmuch easier, for example, by using i-phone, satellite image, etc. As those collected data are not obtained by well-designed probability sampling, inevitably, they may not represent the general population. As a consequence, there probably exists a systematic bias. In the survey sampling literature, how to combine survey sampling data with non probability sampling data has also got very popular (Chen et al., 2020). Without bias correction, most existing methods may produce biased results if the common distribution assumption is violated. One has to be careful to assess the impartiality before data integration. Before we respond to the common concern by the reviewers on the heterogeneity among different studies, we first outline the possible distributional shifts or changes in each source data. In themachine learning literature, the concepts of covariate shift, label shift, and transfer learning have been widely used (QuiñoneroCandela et al., 2009). We briefly highlight those concepts in terms of statistical joint density or conditional density. Covariate shift: Let Y and X be, respectively, the outcome and a vector of covariates in Statistic terminology, or a label variable and a vector of features in Machine Learning Languish. Suppose we have two data-sets: