Abstract:
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We propose a randomized hypothesis test for high dimension low sample size (HDLSS) data. Resampling methods such as the permutation and sign-change tests are popular in hypothesis testing with HDLSS data, especially when the null distribution of a test statistic is not readily available. However, when the hypotheses are regarding the disposition of the data such as in clustering analysis, permutation-based tests are not applicable since they cannot change the data configuration. In this work, we propose a new randomization method called the random dual rotation (RDR) by considering a family of random matrices whose distribution is invariant under a specific group of transformation. The randomized test based on RDR generalizes permutation and sign-change tests, and it can change the configuration of the data while preserving the covariance structure. We provide a theoretical construction of the RDR test by studying the invariance measures on Stiefel and Grassmann manifolds. The proposed idea is applied to outlier detection, cluster verification and two sample comparison problems.
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