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

Abstract

In several applications, information is drawn from quali- tative variables. In such cases, measures of central tendency and dis- persion may be highly inappropriate. Variability for categorical data can be correctly quantied by the so-called diversity measures. These measures can be modied 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 hetero- geneity 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 innite 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 trans- formed 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.

Evaluation of food purchasing in the Brazilian School Feeding Programme: feasibility of the requirements and recommendations

“…kg, unit, can, pack), was converted into grams. For this purpose, the table designed by Pinheiro et al (31) was used to evaluate food in common household measures, and for foods not available in this table, Embrapa's and supermarkets' websites were used as sources of information (32) . Subsequently, using correction factors, the inedible fraction of foods was excluded (e.g.…”