Characterization of Generalized Alpha-Beta Divergence and Associated Entropy Measures

Minimum divergence estimators provide a natural framework for robust (parametric) statistical inference.

Useful properties of several such divergence measures, including, the Hellinger distance, the power divergence, the density power divergence, the logarithmic density power divergence, etc., have been established in the literature; many of them lead to estimators with high statistical efficiency, sometimes even full asymptotic efficiency.

The notable success of these divergences as tools of parametric inference motivates us to explore possible extensions of the alpha-beta divergence family, leading to a superfamily of divergence measures called the ``generalized alpha-beta (GAB) divergences''.

This family contains all the aforementioned popular divergence measures as special cases, and additionally provides opportunities to discover new and novel classes of divergences that generate estimators having strong robustness properties without allowing a significant drop in statistical efficiency in various applications.

In this paper, we provide the necessary and sufficient conditions for the validity of these generalized divergence measures that enable us to employ them for improved statistical inference.

We also show various characterizing properties like duality, inversion, semi-continuity, etc., for the general class of GAB divergences.

A discussion on the entropy measure derived from this general family and its properties are also presented along with the associated maximum entropy principle.

The class of GAB divergences provide a delicate balance between local and global robustness, and this is illustrated by two examples of robust parameter estimation under the Geometric and the normal scale models.