MATHEMATICAL STATISTICS FOR DATA ANALYSIS IN HIGH DIMENSION, LOW SAMPLE SIZE CONTEXT AND ITS APPLICATIONS

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We developed the high-dimension asymptotic theory for High Dimension, Low Sample Size (HDLSS) datasets under a general setup such as non-Gaussian distributions. We found several geometric structures of HDLSS datasets. We showed that the naive PCA is inconsistent in the HDLSS context. We proposed effective inference methods called (1) the noise-reduction methodology, and (2) the cross-data-matrix methodology. By using those methodologies, we gave consistent estimation for intrinsic dimensionality, eigenvalues, their limiting distributions, PC directions and PC scores in the HDLSS context. We applied those methodologies to the discriminant analysis and the cluster analysis in HDLSS data situations from a microarray study of prostate cancer.