  Fusion learning refers to synthesizing inferences from multiple sources or
studies to provide more effective inference and prediction than from any
individual source or study alone. Most existing methods for synthesizing
inferences rely on parametric model assumptions, such as normality, which often
do not hold in practice. In this paper, we propose a general nonparametric
fusion learning framework for synthesizing inferences of the target parameter
from multiple sources. The main tool underlying the proposed framework is the
notion of depth confidence distribution (depth-CD), which is also developed in
this paper. Broadly speaking, a depth-CD is a data-driven nonparametric summary
distribution of inferential information for the target parameter. We show that
a depth-CD is a useful inferential tool and, moreover, is an omnibus form of
confidence regions (or p-values), whose contours of level sets shrink toward
the true parameter value. The proposed fusion learning approach combines
depth-CDs from the individual studies, with each depth-CD constructed by
nonparametric bootstrap and data depth. This approach is shown to be efficient,
general and robust. Specifically, it achieves high-order accuracy and Bahadur
efficiency under suitably chosen combining elements. It allows the model or
inference structure to be different among individual studies. And it readily
adapts to heterogeneous studies with a broad range of complex and irregular
settings. This property enables it to utilize indirect evidence from incomplete
studies to gain efficiency in the overall inference. The advantages of the
proposed approach are demonstrated simulations and in a Federal Aviation
Administration (FAA) study of aircraft landing performance.
