The approximate Bahadur slope of the Chernoff-Lehmann $\chi^2$-test-of-fit to a scale-location family on $R^k$ is computed. The goal is to select cells (whose number is independent of sample size) to maximize this slope. The supremum is found and is shown to be a maximum only in trivial cases. If the $\sup$ is finite there is always a best selection for a fixed number of cells. Equally likely cells are shown to be admissible when the alternative is large.
"Cell Selection in the Chernoff-Lehmann Chi-Square Statistic." Ann. Statist. 4 (2) 375 - 383, March, 1976. https://doi.org/10.1214/aos/1176343413