Bahadur and Raghavachari have formulated the likelihood ratio method of finding the exact slope of a sequence of test statistics which does not require explicit estimation of large deviation probabilities. The method is described herein, and readily verifiable conditions under which it may be invoked are given. The method is then used to find the exact slopes of certain sequences of test statistics arising in multivariate analysis.
"Exact Slopes of Certain Multivariate Tests of Hypotheses." Ann. Statist. 6 (3) 546 - 558, May, 1978. https://doi.org/10.1214/aos/1176344199