  Minimum divergence methods are popular tools in a variety of statistical
applications. We consider tubular model adequacy tests, and demonstrate that
the new divergences that are generated in the process are very useful in robust
statistical inference. In particular we show that the family of $S$-divergences
can be alternatively developed using the tubular model adequacy tests; a
further application of the paradigm generates a larger superfamily of
divergences. We describe the properties of this larger class and its potential
applications in robust inference. Along the way, the failure of the first order
influence function analysis in capturing the robustness of these procedures is
also established.
