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

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.