Bounding measures of genetic similarity and diversity using majorization (Q667701)

To further advance the study of mathematical properties of population-genetic statistics describing similarity and diversity of alleles in a population, the authors investigate an application of the theory of majorization to these statistics. They exploit the majorization theory to study the relationship of the frequency of the most frequent allele to various homozygosity-related statistics as well as to the Shannon-Wearer entropy statistic for genetic diversity. The method introduced by the authors provide simpler derivations of results reported by \textit{N. A. Rosenberg} and \textit{M. Jakobsson} [``The relationship between homozygosity and the frequency of the most frequent allele'', Genetics 179, 2027--2036 (2008; \url{doi:10.1534/genetics.107.084772})] and \textit{S. B. Reddy} and \textit{N. A. Rosenberg} [J. Math. Biol. 64, No. 1--2, 87--108 (2012; Zbl 1284.92058)]; these derivations naturally lead to consider a larger family of homozygosity-related statistics called $\alpha$-homozygosities, where extreme values occur at the same allele frequency vectors as the standard homozygosity. The constraints are illustrated on the statistics using data from human populations.