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A Generalized Relative (α, β)-Entropy: Geometric Properties and Applications to Robust Statistical Inference.

Abhik Ghosh¹, Ayanendranath Basu¹

¹Indian Statistical Institute, Kolkata 700108, India.

Entropy (Basel, Switzerland)

|December 3, 2020

Summary

This summary is machine-generated.

This study introduces general (α, β)-entropy, a new measure generalizing relative α-entropy and logarithmic super divergence (LSD). It establishes the existence and uniqueness of minimum LSD estimators for robust statistical inference.

Keywords:

generalized renyi entropy logarithmic super divergence minimum divergence inference relative entropy robustness

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Area of Science:

Information Theory and Statistics
Mathematical Statistics
Information Geometry

Background:

Relative entropies and divergence measures are vital in information theory and statistics, particularly for robust inference via the minimum divergence principle.
Existing robust methods include ϕ-divergence, density power divergence, logarithmic density power divergence, and logarithmic super divergence (LSD).

Purpose of the Study:

To present an information-theoretic formulation of logarithmic super divergence (LSD) as a two-parameter general (α, β)-entropy.
To explore the geometric properties, relationships with other entropies/divergences, and applications of the general (α, β)-entropy.

Main Methods:

Formulation of general (α, β)-entropy as a two-parameter generalization of relative α-entropy.
Analysis of geometric properties: continuity, convexity, and an extended Pythagorean relation.
Derivation of conditions for the existence and uniqueness of forward and reverse projections.

Main Results:

The general (α, β)-entropy is introduced, offering a unified framework that includes a two-parameter extension of Renyi entropy.
Continuity and convexity of the general (α, β)-entropy are proven, along with an extended Pythagorean relation.
Sufficient conditions for unique projections are established, proving the existence and uniqueness of minimum LSD estimators.

Conclusions:

The general (α, β)-entropy provides a flexible tool with significant geometric properties.
The established existence and uniqueness of minimum LSD estimators open new avenues for robust statistical inference.
Potential applications in robust parameter estimation and hypothesis testing are highlighted, with numerical illustrations for binomial parameter estimation.