In this paper we consider generalized chi-square goodness-of-fit tests based on increasingly finer partitions (as the sample size increases) for models with location-scale nuisance parameters. The asymptotic distributions are derived both under the null hypothesis and under local alternatives, obtained by taking contamination families of densities between the null hypothesis and fixed alternative hypotheses. If the number of random cells increases to infinity, the Rao-Robson-Nikulin test statistic is shown to be superior to the Watson-Roy and Dzhaparidze-Nikulin statistics. Conditions are derived under which it is optimal to let the number of classes tend to infinity.
"Generalized Chi-Square Goodness-of-Fit Tests for Location-Scale Models when the Number of Classes Tends to Infinity." Ann. Statist. 17 (3) 1285 - 1300, September, 1989. https://doi.org/10.1214/aos/1176347269