Covariance Representations and Coherent Measures for Some Entropies

Area of Science:

• Information Theory
• Statistical Modeling
• Probability Distributions

Background:

• Entropies are crucial for quantifying uncertainty and risk in statistical analysis.
• Existing entropy measures may not fully capture tail-dependent risks.
• Covariance and Choquet integral representations provide frameworks for analyzing entropy properties.

Purpose of the Study:

• To derive covariance and Choquet integral representations for specific entropies.
• To establish upper bounds and discuss coherent properties of these entropies.
• To propose novel tail-based entropy measures for enhanced risk assessment.

Main Methods:

• Obtained covariance and Choquet integral representations for selected entropies.
• Derived upper bounds for these entropies.
• Introduced tail-based cumulative residual Tsallis entropy of order α (TCRTE) and tail-based right-tail deviation (TRTD).
• Defined shortfall of cumulative residual Tsallis (CRTES) and shortfall of right-tail deviation entropy (RTDS).

Main Results:

• Established equivalent results for the newly defined shortfall entropies.
• Demonstrated the application of CRTES through simulations for elliptical, inverse Gaussian, gamma, and beta distributions.
• Provided upper bounds and discussed the coherent properties of the discussed entropies.

Conclusions:

• The proposed tail-based entropy measures (TCRTE, TRTD, CRTES, RTDS) offer valuable tools for risk analysis, particularly in capturing tail behaviors.
• The covariance and Choquet integral representations provide theoretical underpinnings for these entropy measures.
• Simulations confirm the utility of CRTES for various continuous probability distributions.