We present a unified large-sample theory of general chi-squared tests of fit under composite null hypotheses and Pitman alternatives. The statistics are quadratic forms in the standardized cell frequencies, and we allow random cells, $k$-variate observations from not necessarily continuous distributions, and quite general estimates of unknown parameters. Generalizations of published results on a number of specific chi-squared tests follow.
"Unified Large-Sample Theory of General Chi-Squared Statistics for Tests of Fit." Ann. Statist. 3 (3) 599 - 616, May, 1975. https://doi.org/10.1214/aos/1176343125