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The homogeneity, or more generally, the similarity between source domains anda target domain seems to be essential to a positive transfer learning. Inpractice, however, the similarity condition is difficult to check and is oftenviolated. In this paper, instead of the popularly used similarity condition, aseeming similarity is introduced, which is defined by a non-orthogonalitytogether with a smoothness. Such a condition is naturally satisfied undercommon situations and even implies the dissimilarity in some sense. Based onthe seeming similarity together with an $L_2$-adjustment, a source-functionweighted-transfer learning estimation (sw-TLE) is constructed. Bysource-function weighting, an adaptive transfer learning is achieved in thesense that it is applied to similar and dissimilar scenarios with a relativelyhigh estimation efficiency. Particularly, under the case with homogenous sourceand target models, the sw-TLE even can be competitive with the full dataestimator. The hidden relationship between the source-function weightingestimator and the James-Stein estimator is established as well, which revealsthe structural reasonability of our methodology. Moreover, the strategy doesapply to nonparametric and semiparametric models. The comprehensive simulationstudies and real data analysis can illustrate that the new strategy issignificantly better than the competitors.

Source-Function Weighted-Transfer Learning for Nonparametric Regression with Seemingly Similar Sources