We consider general chi-square goodness-of-fit test statistics for randomly censored data--call these generalized Pearson statistics--which are nonnegative definite quadratic forms in the cell frequencies obtained from the product-limit estimator, allowing random cells and general estimators of nuisance parameters. This class of statistics generalizes the class studied by Moore and Spruill in the no censoring case. The large sample behavior of these statistics under the null hypothesis and local alternatives is presented. The chi-square type statistics based on the observed cell frequencies obtained from the product-limit estimator are members of this class for which the quadratic form is selected to produce a chi-square asymptotic null distribution. The generalized Pearson statistic and the statistic by Akritas for a simple hypothesis are compared on the basis of asymptotic relative Pitman efficiency. It is shown that neither statistic dominates the other. The efficiencies are shown to depend on the degree of censoring and the number of cells. For heavily censored data, the Akritas statistic is superior to the generalized Pearson statistic. In the uncensored case, the Akritas statistic, which does not reduce to the Pearson statistic, is not as good as the Pearson statistic in the sense of Pitman efficiency.
"Chi-Square Goodness-of-Fit Tests for Randomly Censored Data." Ann. Statist. 21 (3) 1621 - 1639, September, 1993. https://doi.org/10.1214/aos/1176349275