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Fusion learning refers to synthesizing inferences from multiple sources orstudies to provide more effective inference and prediction than from anyindividual source or study alone. Most existing methods for synthesizinginferences rely on parametric model assumptions, such as normality, which oftendo not hold in practice. In this paper, we propose a general nonparametricfusion learning framework for synthesizing inferences of the target parameterfrom multiple sources. The main tool underlying the proposed framework is thenotion of depth confidence distribution (depth-CD), which is also developed inthis paper. Broadly speaking, a depth-CD is a data-driven nonparametric summarydistribution of inferential information for the target parameter. We show thata depth-CD is a useful inferential tool and, moreover, is an omnibus form ofconfidence regions (or p-values), whose contours of level sets shrink towardthe true parameter value. The proposed fusion learning approach combinesdepth-CDs from the individual studies, with each depth-CD constructed bynonparametric bootstrap and data depth. This approach is shown to be efficient,general and robust. Specifically, it achieves high-order accuracy and Bahadurefficiency under suitably chosen combining elements. It allows the model orinference structure to be different among individual studies. And it readilyadapts to heterogeneous studies with a broad range of complex and irregularsettings. This property enables it to utilize indirect evidence from incompletestudies to gain efficiency in the overall inference. The advantages of theproposed approach are demonstrated simulations and in a Federal AviationAdministration (FAA) study of aircraft landing performance.

Nonparametric fusion learning: synthesize inferences from diverse sources using depth confidence distribution