On the super-additivity and estimation biases of quantile contributions

• Estimating concentration (inequality or dispersion) or other statistical properties (such as severity of violent conflicts) from top quantile contributions is inconsistent under aggregation.
• The measure increases with the size of the total population and converges very slowly.
• The bias is more acute at fatter tails, lower tail exponent alpha and smaller centile.
• The weighted average of measures for AA and BB will be ≤≤ than that from A∪BA∪B.
• The effect is exacerbated under mixing distributions (stochastic tail exponent).

Sample measures of top centile contributions to the total (concentration) are downward biased, unstable estimators, extremely sensitive to both sample and population size and concave in accounting for large deviations. It makes them particularly unfit in domains with power law tails, especially for low values of the exponent. These estimators can vary over time and increase with the population size, thus providing the illusion of structural changes in concentration. They are also inconsistent under aggregation and mixing distributions, as the weighted average of concentration measures for AA and BB will tend to be lower than that from A∪BA∪B. In addition, it can be shown that under such fat tails, increases in the total sum need to be accompanied by increased sample size of the concentration measurement. We examine the estimation superadditivity and bias under homogeneous and mixed distributions.