Decomposability of high-dimensional diversity measures: Quasi--statistics, martingales and nonstandard asymptotics |
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Authors: | Aluísio Pinheiro Pranab Kumar Sen Hildete Prisco Pinheiro |
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Institution: | aDepartment of Statistics, IMECC, University of Campinas, Brazil;bDepartment of Biostatistics, UNC-Chapel Hill, NC, United States;cDepartment of Statistics and Operations Research, UNC-Chapel Hill, NC, United States |
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Abstract: | In analyses of complex diversity, especially that arising in genetics, genomics, ecology and other high-dimensional (and sometimes low-sample-size) data models, typically subgroup decomposability (analogous to ANOVA decomposability) arises. For group divergence of diversity measures in a high-dimension low-sample-size scenario, it is shown that Hamming distance type statistics lead to a general class of quasi-U-statistics having, under the hypothesis of homogeneity, a martingale (array) property, providing a key to the study of general (nonstandard) asymptotics. Neither the stochastic independence nor homogeneity of the marginal probability laws plays a basic role. A genomic MANOVA model is presented as an illustration. |
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Keywords: | Categorical Data Dependence DNA Genomics Hamming distance Orthogonal system Permutation measure Second-order asymptotics Second-order decomposability |
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