Pearson-type goodness-of-fit tests: The univariate case

Chi-squared tests are proposed for testing goodness of fit when the data may or may not be subject to random censoring. Both the simple and compositive null hypotheses are considered. For the simple null hypothesis the resulting test statistic in the uncensored case differs from Pearson’s statistic only in that the expected frequency for cell A_jis taken to be (Equation presented), where (Equation presented) is the empirical distribution function, instead of n ∫_AjdF_o; this results in k df instead of the usual k − 1. The theoretical justification lies in the behavior of the corresponding modified version of the usual empirical process: It converges weakly to a Brownian motion process instead of a tied-down Brownian motion. Under censoring, the tests require no restrictive assumptions on the censoring distribution and the statistic has the same simple form. For the composite null hypothesis the tests’ development is based on the weak convergence of the modified empirical process when parameters are estimated. The raw-data maximum likelihood estimator and the minimum chi-squared estimator are considered in estimating unknown parameters. In both cases the form of the test statistic is the same as the usual chi-squared statistic, but again it has an additional degree of freedom. The additional degree of freedom results in simple tests based on only one cell. A number of examples are presented; the tests for the negative exponential, the Weibull, the Pareto, and the one-parameter gamma families are the simplest available tests under random censoring. Results of a simulation study and an analysis of a real data set are presented.