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K. Hron, M. Templ, P. Filzmoser:
"Estimation of a proportion in survey sampling using the logratio approach";
Metrika, 76 (2013), 6; 799 - 818.

The estimation of a mean of a proportion is a frequent task in statistical survey analysis, and often such ratios are estimated from compositions such as income components, wage components, tax components, etc. In practice, the weighted arithmetic mean is regularly used to estimate the center of the data. However, this estimator is not appropriate if the ratios are estimated from compositions, because the sample space of compositional data is the simplex and not the usual Euclidean space. We demonstrate that the weighted geometric mean is useful for this purpose. Even for different sampling designs, the weighted geometric mean shows excellent behavior.

Survey sampling Proportions Compositional data Logratio analysis

http://dx.doi.org/10.1007/s00184-012-0416-6

http://link.springer.com/article/10.1007/s00184-012-0416-6/fulltext.html#Sec1