  A full parametric and linear specification may be insufficient to capture
complicated patterns in studies exploring complex features, such as those
investigating age-related changes in brain functional abilities. Alternatively,
a partially linear model (PLM) consisting of both parametric and non-parametric
elements may have a better fit. This model has been widely applied in
economics, environmental science, and biomedical studies. In this paper, we
introduce a novel statistical inference framework that equips PLM with high
estimation efficiency by effectively synthesizing summary information from
external data into the main analysis. Such an integrative scheme is versatile
in assimilating various types of reduced models from the external study. The
proposed method is shown to be theoretically valid and numerically convenient,
and it ensures a high-efficiency gain compared to classic methods in PLM. Our
method is further validated using two data applications by evaluating the risk
factors of brain imaging measures and blood pressure.
