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Decomposability of high-dimensional diversity measures: Quasi-U-statistics, martingales and nonstandard asymptotics

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  • Pinheiro, Aluísio
  • Sen, Pranab Kumar
  • Pinheiro, Hildete Prisco

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.

Suggested Citation

  • Pinheiro, Aluísio & Sen, Pranab Kumar & Pinheiro, Hildete Prisco, 2009. "Decomposability of high-dimensional diversity measures: Quasi-U-statistics, martingales and nonstandard asymptotics," Journal of Multivariate Analysis, Elsevier, vol. 100(8), pages 1645-1656, September.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:8:p:1645-1656
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    References listed on IDEAS

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    1. Sen, Pranab K. & Tsai, Ming-Tien & Jou, Yuh-Shan, 2007. "High-Dimension, LowSample Size Perspectives in Constrained Statistical Inference: The SARSCoV RNA Genome in Illustration," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 686-694, June.
    2. Tzeng J-Y. & Byerley W. & Devlin B. & Roeder K. & Wasserman L., 2003. "Outlier Detection and False Discovery Rates for Whole-Genome DNA Matching," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 236-246, January.
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    Citations

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    Cited by:

    1. Aluísio Pinheiro & Pranab Sen & Hildete Pinheiro, 2011. "A class of asymptotically normal degenerate quasi U-statistics," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(6), pages 1165-1182, December.
    2. Juvêncio Nobre & Julio Singer & Pranab Sen, 2013. "U-tests for variance components in linear mixed models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(4), pages 580-605, November.
    3. Chen, Fei & Li, Zaixing & Shi, Lei & Zhu, Lixing, 2015. "Inference for mixed models of ANOVA type with high-dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 382-401.
    4. Soukarieh, Inass & Bouzebda, Salim, 2023. "Renewal type bootstrap for increasing degree U-process of a Markov chain," Journal of Multivariate Analysis, Elsevier, vol. 195(C).
    5. Rauf Ahmad, M. & Pavlenko, Tatjana, 2018. "A U-classifier for high-dimensional data under non-normality," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 269-283.
    6. Hildete P. Pinheiro & Rafael P. Maia & Eufrásio A. Lima Neto & Mariana Rodrigues-Motta, 2019. "Zero-one augmented beta and zero-inflated discrete models with heterogeneous dispersion for the analysis of student academic performance," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 28(4), pages 749-767, December.

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