An optimal property of the exact multinomial test and the extended Fisher's exact test

In the goodness-of-fit test of parameters of the multinomial distribution we show that the exact multinomial test is asymptotically equivalent to the likelihood ratio test by using Stirling's formula. In an r×c contingency table, we show that the extended Fisher's exact test conditional on row and column margins for the test of independence is also asymptotically equivalent to the likelihood ratio test. From the Bahadur asymptotic optimality of the likelihood ratio test in both unconditional and conditional cases, we prove that the two exact tests are asymptotically optimal in the sense of Bahadur efficiency.