The Pearson-Fisher $\chi^2$ statistic is asymptotically chi-square under the null hypothesis with $M - m - 1$ degrees of freedom where $M =$ number of cells and $m =$ dimension of parameter. The Chernoff-Lehmann statistic is a weighted sum of chi-squares and the Kambhampati statistic is $\chi^2$ with $M - 1$ degrees of freedom. The approximate Bahadur slopes of the tests based on these statistics are computed. It is shown that the Kambhampati test always dominates the Chernoff-Lehmann and that no such dominance exists between the Pearson-Fisher test and Kambhampati test, or the Pearson-Fisher and Chernoff-Lehmann.
"A Comparison of Chi-Square Goodness-of-Fit Tests Based on Approximate Bahadur Slope." Ann. Statist. 4 (2) 409 - 412, March, 1976. https://doi.org/10.1214/aos/1176343418