Consistency of chi-squared test with varying number of classes

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• 作者：Rui Huang (1) (2) (3)
  Hengjian Cui (1) (2) (3)
  
  1. College of Mathematic and Statistics ; Jishou University ; Jishou ; 416000 ; China
  2. School of Mathematical Sciences ; Beijing Normal University ; Beijing ; 100875 ; China
  3. Department of Statistics ; School of Mathematical Sciences ; Capital Normal University ; Beijing ; 100048 ; China
  
• 关键词：Consistency of chi ; squared test ; goodness of fit test ; varying number of classes
• 刊名：Journal of Systems Science and Complexity
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：28
• 期：2
• 页码：439-450
• 全文大小：277 KB
• 参考文献：1. Pearson, K (1900) On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philosophical Magazine 50: pp. 157-175 CrossRef
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  3. Mann, H B, Wald, A (1942) On the choice of the number of class intervals in the application of the chi-square test. The Annals of Mathematical Statistics 13: pp. 306-317 CrossRef
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  5. Williams, C A (1950) On the choice of the number and width of classes for the chi-square test of goodness of fit. Journal of the American Statistical Association 45: pp. 77-86
  6. Rayner, J C W, Best, D J (1982) The choice of class probabilities and number of classes for the simple X 2 goodness of fit test. The Indian Journal of Statistics 44: pp. 28-38
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  10. G枚tze, F, Tikhomirov, A N (1999) Asymptotic distribution of quadratic forms. The Annals of Probability 27: pp. 1072-1098 CrossRef
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  12. Noel, C, Timothy, R C R (1984) Multinomial goodness of fit tests. Journal of the Royal Statistical Society 46: pp. 440-464
  13. Whittle, P (1964) On the convergence to normality of quadratic forms in independent variables. Theory of Probability and Its Applications 9: pp. 103-108 CrossRef
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Systems Theory and Control
  Applied Mathematics and Computational Methods of Engineering
  Operations Research/Decision Theory
  Probability Theory and Stochastic Processes
  
• 出版者：Academy of Mathematics and Systems Science, Chinese Academy of Sciences, co-published with Springer
• ISSN：1559-7067

文摘

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.