How to measure inequality when data is incomplete?

Imagine wanting to understand how fairly money is distributed among a group of friends, but someone only tells you: 'Marco has between 10 and 20 euros' or 'The group has 100 euros in total.' Difficult, right? This is exactly the problem scientists face when studying real-world wealth. Often, we don't have precise data for every single person, but only 'groups' or ranges of values.

A new mathematical study has found a clever way to solve this mystery. Scientists have developed a system capable of calculating 'bounds.' Instead of providing a single number that might be wrong, the system says: 'The inequality is definitely between this minimum value and this maximum.' It's like saying you don't know the exact height of a wall, but you know it's between 2 and 3 meters.

This method works with famous indices like the Gini coefficient, which measures how wealth is distributed. By using advanced mathematical calculations, researchers have managed to transform infinite and extremely complex problems into fast calculations that a computer can solve in an instant. This is crucial for truly understanding how wealth changes over time, even when historical data is unclear or grouped in tables.

The core of this research lies in 'partial identification.' In statistics, when data is incomplete (interval observations or grouped data), it is impossible to determine a single point value for a parameter, such as the Gini coefficient. The proposed framework uses Schur-convexity theory to characterize the distributions that maximize or minimize such indices. The authors demonstrated that, for a wide class of inequality measures, the extreme bounds are attained by distributions with finite support, transforming an optimization problem in an infinite-dimensional space into a much more manageable finite-dimensional optimization problem. Furthermore, for indices that can be expressed as linear-fractional ratios (such as quantile ratios), the problem is recast as linear or quadratic programming. This allows for the rapid computation of numerically precise bounds. Regarding inference, the method employs a uniform directional delta method and bootstrap techniques to estimate standard errors, ensuring that the estimated bounds are statistically reliable even with finite samples. This approach was successfully tested on real wealth data (ELSA) and historical U.S. income tables, demonstrating great practical utility for modern economic analysis.