The Characterization of the Symmetric Lorenz Curves by Doubly Truncated Mean Function

Abstract

We characterize symmetric Lorenz curves by the relation m(x, μ²/x) = μ (where μ =E(X) and m(x, y) = E(X | x ≤ X ≤ y) is the doubly truncated mean function). We establish that the points of the r.v. which generate the symmetric points on the Lorenz curve are x and μ²/x, and that all the distribution functions defined on the same support which are generators of the symmetric Lorenz curves have the same mean. We obtain the conditions under which doubly truncated distributions generate symmetrical Lorenz curves.

Keywords:

• Characterization
• Lorenz curve
• Symmetry

Mathematics Subject Classification:

• Primary 05C38, 15A15
• Secondary 05A15, 15A18

Acknowledgment

The authors would like to express their gratitude to the referee for his constructive comments and valuable suggestions, which considerably improved the content and the presentation of the article.