Maximization of Mathai's Entropy under the Constraints of Generalized Gini and Gini mean difference indices and its Applications in Insurance

TL;DR

This paper maximizes Mathai's entropy under constraints involving Gini indices and mean, deriving distributions applicable to insurance loss data, bridging concepts from physics, economics, and statistics.

Contribution

It introduces a novel approach to maximize Mathai's entropy with Gini-based constraints and applies it to real insurance data for improved modeling.

Findings

Maximum entropy distributions fit earthquake insurance loss ratios well.

The approach outperforms some standard distributions in modeling insurance data.

The method links entropy maximization with economic inequality measures.

Abstract

Statistical Physics, Diffusion Entropy Analysis and Information Theory commonly use Mathai's entropy which measures the randomness of probability laws, whereas welfare economics and the Social Sciences commonly use Gini index which measures the evenness of probability laws. Motivated by the principle of maximal entropy, we explore the maximization of Mathai's entropy subject to the conditions in the following scenarios: (i) the conditions of a density function and fixed mean; (ii) the conditions of a density function and fixed Generalized Gini index. We also maximizes the Mathai's entropy subject to the constraints of a given Gini mean difference index and the conditions of a density function. The obtained maximum entropy distribution is fitted to the loss ratios (yearly data) for earthquake insurance in California from 1971 through 1994 and its performance with some one-parameter distributions are compared.