Decomposability of High-Dimensional Diversity Measures: Quasi U-Statistics, Martingales and Nonstandard Asymptotics

In complex diversity analysis, specially arising in genetics, genomics, ecology and other high-dimensional (and sometimes low sample size) data models, typically sub\-group-decomposability (analogous to ANOVA decomposability) arises. In 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 a martingale (array) property, providing key to the study of general (nonstandard) asymptotics. Neither the stochastic independence nor homogeneity of the marginal probability laws play a basic role. A genomic MANOVA model is presented as an illustration.