  It is proved that the information divergence statistic is infinitely more
Bahadur efficient than the power divergence statistics of the orders $\alpha
>1$ as long as the sequence of alternatives is contiguous with respect to the
sequence of null-hypotheses and the the number of observations per bin
increases to infinity is not very slow. This improves the former result in
Harremo\"es and Vajda (2008) where the the sequence of null-hypotheses was
assumed to be uniform and the restrictions on on the numbers of observations
per bin were sharper. Moreover, this paper evaluates also the Bahadur
efficiency of the power divergence statistics of the remaining positive orders
$0< \alpha \leq 1.$ The statistics of these orders are mutually
Bahadur-comparable and all of them are more Bahadur efficient than the
statistics of the orders $\alpha > 1.$ A detailed discussion of the technical
definitions and conditions is given, some unclear points are resolved, and the
results are illustrated by examples.
