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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., Comment: 47 pages, 10 figures

Statistical depth measures the centrality of a point with respect to a given distribution or data cloud. It provides a natural center-outward ordering of multivariate data points and yields a systematic nonparametric multivariate analysis scheme. In particular, the half-space depth is shown to have many desirable properties and broad applicability. However, the empirical half-space depth is zero outside the convex hull of the data. This property has rendered the empirical half-space depth useless outside the data cloud, and limited its utility in applications where the extreme outlying probability mass is the focal point, such as in classification problems and control charts with very small false alarm rates. To address this issue, we apply extreme value statistics to refine the empirical half-space depth in "the tail." This provides an important linkage between data depth, which is useful for inference on centrality, and extreme value statistics, which is useful for inference on extremity. The refined empirical half-space depth can thus extend all its utilities beyond the data cloud, and hence broaden greatly its applicability. The refined estimator is shown to have substantially improved upon the empirical estimator in theory and simulations. The benefit of this improvement is also demonstrated through the applications in classification and statistical process control., Comment: Published at http://dx.doi.org/10.1214/15-AOS1359 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

External information, such as prior information or expert opinions, can play an important role in the design, analysis and interpretation of clinical trials. However, little attention has been devoted thus far to incorporating external information in clinical trials with binary outcomes, perhaps due to the perception that binary outcomes can be treated as normally-distributed outcomes by using normal approximations. In this paper we show that these two types of clinical trials could behave differently, and that special care is needed for the analysis of clinical trials with binary outcomes. In particular, we first examine a simple but commonly used univariate Bayesian approach and observe a technical flaw. We then study the full Bayesian approach using different beta priors and a new frequentist approach based on the notion of confidence distribution (CD). These approaches are illustrated and compared using data from clinical studies and simulations. The full Bayesian approach is theoretically sound, but surprisingly, under skewed prior distributions, the estimate derived from the marginal posterior distribution may not fall between those from the marginal prior and the likelihood of clinical trial data. This counterintuitive phenomenon, which we call the "discrepant posterior phenomenon," does not occur in the CD approach. The CD approach is also computationally simpler and can be applied directly to any prior distribution, symmetric or skewed., Comment: Published in at http://dx.doi.org/10.1214/12-AOAS585 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

This paper introduces and studies multivariate spacings. The spacings are developed using the order statistics derived from data depth. Specifically, the spacing between two consecutive order statistics is the region which bridges the two order statistics, in the sense that the region contains all the points whose depth values fall between the depth values of the two consecutive order statistics. These multivariate spacings can be viewed as a data-driven realization of the so-called ``statistically equivalent blocks.'' These spacings assume a form of center-outward layers of ``shells'' (``rings'' in the two-dimensional case), where the shapes of the shells follow closely the underlying probabilistic geometry. The properties and applications of these spacings are studied. In particular, the spacings are used to construct tolerance regions. The construction of tolerance regions is nonparametric and completely data driven, and the resulting tolerance region reflects the true geometry of the underlying distribution. This is different from most existing approaches which require that the shape of the tolerance region be specified in advance. The proposed tolerance regions are shown to meet the prescribed specifications, in terms of $\beta$-content and $\beta$-expectation. They are also asymptotically minimal under elliptical distributions. Finally, a simulation and comparison study on the proposed tolerance regions is presented., Comment: Published in at http://dx.doi.org/10.1214/07-AOS505 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)