Moore (1971) derives the limiting distribution of chi-square statistics for testing goodness of fit to k-variate parametric families, where the cell boundaries are allowed to be functions of the estimated parameter values. The only point at which random cells, multivariate observations etc. require a deviation from methods of proof given by Cramer (1946) is in the proof of the asymptotic negligibility of the remainder terms. Attention is drawn to an alternative proof of this asymptotic negligibility, which turns out to be an immediate consequence of a modification of Lemma 1 by Bahadur (1966) in more dimensions.
"A Note on Chi-Square Statistics with Random Cell Boundaries." Ann. Statist. 3 (4) 965 - 968, July, 1975. https://doi.org/10.1214/aos/1176343198