Bahadur deficiency of likelihood ratio tests in exponential families

For many testing problems several different tests may have optimal exact Bahadur slope. The introduction of Bahadur deficiency provides further information about the performance of such tests. Roughly speaking a sequence of tests is deficient in the sense of Bahadur of order (hn) at a fixed alternative [theta] if the additional number of observations necessary to obtain the same power as the optimal test at [theta] is of order (hn) as the level of significance tends to zero. In this paper it is shown that in typical testing problems in multivariate exponential families the LR test is deficient in the sense of Bahadur of order (log n).