Consistency of chi-squared test with varying number of classes

Abstract

The classical chi-squared goodness of fit test assumes the number of classes is fixed, meanwhile the test statistic has a limiting chi-square distribution under the null hypothesis. It is well known that the number of classes varying with sample size in the test has attached more and more attention. However, in this situation, there is not theoretical results for the asymptotic property of such chi-squared test statistic. This paper proves the consistency of chi-squared test with varying number of classes under some conditions. Meanwhile, the authors also give a convergence rate of Kolmogorov-Simirnov distance between the test statistic and corresponding chi-square distributed random variable. In addition, a real example and simulation results validate the reasonability of theoretical result and the superiority of chi-squared test with varying number of classes.