We introduce a broad class of normalized scale-invariant indices (NSIIs) generated by measurable symmetric homogeneous functions and encompassing several well-known measures, including the Gini coefficient, generalized Gini indices, entropy-based measures, and variability indices. Explicit expressions are obtained for these indices under Gamma populations. Exploiting the independence between the total sum and the associated Dirichlet proportions, we derive a simple unbiased estimator based on a U-statistic. The resulting estimator is shown to be unbiased for any NSII in the proposed class when the underlying population follows a Gamma distribution. Several examples are provided to illustrate the general theory. A Monte Carlo simulation study is carried out that shows the good performance of the unbiased estimator in several scenarios of index choices. We evaluate the effectiveness of the NSIIs and their estimates in modeling a real-world dataset related to gross domestic product per capita in the Americas.

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