Quasi U-statistics of infinite order and applications to the subgroup decomposition of some diversity measures

In several applications, information is drawn from qualitative variables. In such cases, measures of central tendency and dispersion may be highly inappropriate. Variability for categorical data can be correctly quantified by the so-called diversity measures. These measures can be modified to quantify heterogeneity between groups (or subpopulations). Pinheiro et al. (2005) shows that Hamming distance can be employed in such way and the resulting estimator of heterogeneity between populations will be asymptotically normal under mild regularity conditions. Pinheiro et al. (2009) proposes a class of weighted U -statistics based on degenerate kernels of degree 2, called quasi U -statistics, with the property of asymptotic normality under suitable conditions. This is generalized to kernels of degree m by Pinheiro et al. (2011). In this work we generalize this class to an infinite order degenerate kernel. We then use this powerful tools and the reverse martingale nature of U -statistics to study the asymptotic behavior of a collection of transformed classic diversity measures. We are able to estimate them in a common framework instead of the usual individualized estimation procedures. MSC 2000: primary 62G10; secondary 62G20, 92D20.