Tests of exchangeability for high-dimensional data

Abstract Exchangeability is an important data structure and, if appropriately discovered, may help in discernible dimension reduction. This aspect carries special significance for high-dimensional data. For certain data sets, such as when same units are repeatedly observed, not only across all time points, but further under all experimental conditions, exchangeability is strongly expected, and a test to determine its presence is desired. For example, the same patients might be observed across several time points, under both drug and placebo. In another setting, expressions from normal and tumor cells from the same individuals may be evaluated. The notion of exchangeability in such cases implies that the distinction of different conditions is of no significance, or that the sub-vectors under different conditions are exchangeable. The aim of this article is to present tests for multivariate exchangeability when the distributions may not necessarily be normal and the dimensions of the vectors, even of sub-vectors, may exceed the sample size. The test statistics are composed of consistent and computationally highly efficient estimators, and their properties are studied under a few mild assumptions. Extension to two-layer pattern of exchangeability, i.e., when the sub-vectors are further partitioned into sub-sub-vectors, based on an additional variable, is also considered. Simulation results are used to show the accuracy of the proposed tests. Applications on real data examples are demonstrated.