  M-estimators offer simple robust alternatives to the maximum likelihood
estimator. Much of the robustness literature, however, has focused on the
problems of location, location-scale and regression estimation rather than on
estimation of general parameters. The density power divergence (DPD) and the
logarithmic density power divergence (LDPD) measures provide two classes of
competitive M-estimators (obtained from divergences) in general parametric
models which contain the MLE as a special case. In each of these families, the
robustness of the estimator is achieved through a density power down-weighting
of outlying observations. Both the families have proved to be very useful tools
in the area of robust inference. However, the relation and hierarchy between
the minimum distance estimators of the two families are yet to be
comprehensively studied or fully established. Given a particular set of real
data, how does one choose an optimal member from the union of these two classes
of divergences? In this paper, we present a generalized family of divergences
incorporating the above two classes; this family provides a smooth bridge
between the DPD and the LDPD measures. This family helps to clarify and settle
several longstanding issues in the relation between the important families of
DPD and LDPD, apart from being an important tool in different areas of
statistical inference in its own right.
