Describing the concentration of income populations by functional principal component analysis on Lorenz curves

Lorenz curves are widely used in economic studies (inequality, poverty, differentiation, etc.). From a model point of view, such curves can be seen as constrained functional data for which functional principal component analysis (FPCA) could be defined. Although statistically consistent, performing FPCA using the original data can lead to a suboptimal analysis from a mathematical and interpretation point of view. In fact, the family of Lorenz curves lacks very basic (e.g., vectorial) structures and, hence, must be treated with ad hoc methods. This work aims to provide a rigorous mathematical framework via an embedding approach to define a coherent FPCA for Lorenz curves. This approach is used to explore a functional dataset from the Bank of Italy income survey.