The chi-square test, likelihood ratio test, and other goodness of fit tests in a family are shown to have monotonic power functions, and hence are unbiased, for testing a simple hypothesis that all cell probabilities are equal. The chi-square test is also shown to be Type D for this situation.
"Unbiasedness of the Chi-Square, Likelihood Ratio, and Other Goodness of Fit Tests for the Equal Cell Case." Ann. Statist. 3 (4) 959 - 964, July, 1975. https://doi.org/10.1214/aos/1176343197