  Sample measures of top centile contributions to the total (concentration) are
downward biased, unstable estimators, extremely sensitive to sample 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, as shown
in this article, 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 A and B
will tend to be lower than that from A U 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.
