On the Super-Additivity and Estimation Biases of Quantile Contributions

TL;DR

This paper investigates the biases and super-additivity properties of sample-based measures of top contribution concentration, revealing their instability, bias, and sensitivity to sample size, especially in fat-tailed distributions.

Contribution

It provides a detailed analysis of the biases and super-additivity of quantile-based concentration measures under various distributional assumptions.

Findings

Sample concentration measures are downward biased and unstable.

These measures are highly sensitive to sample size and distribution tails.

Aggregation can lead to underestimation of combined concentration.

Abstract

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.