Abstract

Fusion learning refers to synthesizing inferences from multiple sources or studies to make a 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. We propose a general nonparametric fusion learning framework for synthesizing inferences for multiparameters from different studies. The main tool underlying the proposed framework is the new notion of depth confidence distribution (depth-CD), which is developed by combining data depth and confidence distribution. Broadly speaking, a depth-CD is a data-driven nonparametric summary distribution of the available inferential information for a target parameter. We show that a depth-CD is a powerful inferential tool and, moreover, is an omnibus form of confidence regions, 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. The 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 last property enables the approach to use indirect evidence from incomplete studies to gain efficiency for the overall inference. We develop the theoretical support for the proposed approach, and we also illustrate the approach in making combined inference for the common mean vector and correlation coefficient from several studies. The numerical results from simulated studies show the approach to be less biased and more efficient than the traditional approaches in nonnormal settings. The advantages of the approach are also demonstrated in a Federal Aviation Administration study of aircraft landing performance. Supplementary materials for this article are available online.

摘要

融合学习是指综合来自多个来源或研究的推论，以做出比来自任何单独来源或单独研究更有效的推论和预测。大多数现有的推理综合方法都依赖于参数模型假设，例如正态性，而这在实践中通常不成立。我们提出了一个通用的非参数融合学习框架，用于综合来自不同研究的多参数的推论。所提出框架的主要工具是深度置信度分布 (depth-CD)的新概念，它是通过结合数据深度和置信度分布而开发的。从广义上讲，深度 CD是目标参数的可用推理信息的数据驱动的非参数摘要分布。我们表明，深度 CD是一种强大的推理工具，而且是置信区域的综合形式，其水平集的轮廓向真实参数值收缩。所提出的融合学习方法结合了来自各个研究的深度 CD，每个深度CD由非参数自举和数据深度构建。该方法被证明是有效的、通用的和稳健的. Specifically, it achieves high-order accuracy and Bahadur efficiency under suitably chosen combining elements. 它允许模型或推理结构在各个研究中有所不同。而且，它很容易适应具有广泛复杂和不规则设置的异质研究。最后一个属性使该方法能够使用来自不完整研究的间接证据来提高整体推理的效率。我们为所提出的方法提供了理论支持，我们还说明了从几项研究中对共同均值向量和相关系数进行联合推断的方法。模拟研究的数值结果表明，与非正态设置中的传统方法相比，该方法的偏差更小，效率更高。该方法的优点还体现在美国联邦航空管理局对飞机着陆性能的研究。本文的补充材料可在线获取。