A general asymptotic framework for distribution‐free graph‐based two‐sample tests
Testing equality of two multivariate distributions is a classical problem for which many non‐parametric tests have been proposed over the years. Most of the popular two‐sample tests, which are asymptotically distribution free, are based either on geometric graphs constructed by using interpoint distances between the observations (multivariate generalizations of the Wald–Wolfowitz runs test) or on multivariate data depth (generalizations of the Mann–Whitney rank test). The paper introduces a general notion of distribution‐free graph‐based two‐sample tests and provides a unified framework for analysing and comparing their asymptotic properties. The asymptotic (Pitman) efficiency of a general graph‐based test is derived, which includes tests based on geometric graphs, such as the Friedman–Rafsky test, the test based on the K‐nearest‐neighbour graph, the cross‐match test and the generalized edge count test, as well as tests based on multivariate depth functions (the Liu–Singh rank sum statistic). The results show how the combinatorial properties of the underlying graph affect the performance of the associated two‐sample test and can be used to validate and decide which tests to use in practice. Applications of the results are illustrated both on synthetic and on real data sets.
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