  Minimization of suitable statistical distances~(between the data and model
densities) has proved to be a very useful technique in the field of robust
inference. Apart from the class of $\phi$-divergences of \cite{a} and \cite{b},
the Bregman divergence (\cite{c}) has been extensively used for this purpose.
However, since the data density must have a linear presence in the cross
product term of the Bregman divergence involving both the data and model
densities, several useful divergences cannot be captured by the usual Bregman
form. In this respect, we provide an extension of the ordinary Bregman
divergence by considering an exponent of the density function as the argument
rather than the density function itself. We demonstrate that many useful
divergence families, which are not ordinarily Bregman divergences, can be
accommodated within this extended description. Using this formulation, one can
develop many new families of divergences which may be useful in robust
inference. In particular, through an application of this extension, we propose
the new class of the GSB divergence family. We explore the applicability of the
minimum GSB divergence estimator in discrete parametric models. Simulation
studies as well as conforming real data examples are given to demonstrate the
performance of the estimator and to substantiate the theory developed.
