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.

中文翻译：

---

卡类检验与不同类数的一致性

经典的卡方拟合优度检验假设类别数固定，同时在原假设下，检验统计量具有有限的卡方分布。众所周知，测试中随样本大小而变化的类别数量越来越受到关注。但是，在这种情况下，没有关于这种卡方检验统计量的渐近性质的理论结果。本文证明了在一定条件下不同类别的卡方检验的一致性。同时，作者还给出了检验统计量和相应的卡方分布随机变量之间的Kolmogorov-Simirnov距离的收敛速度。此外，