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Consistency of chi-squared test with varying number of classes

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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.

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Author information

Authors and Affiliations

  1. College of Mathematic and Statistics, Jishou University, Jishou, 416000, China

    Rui Huang & Hengjian Cui

  2. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Rui Huang & Hengjian Cui

  3. Department of Statistics, School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    Rui Huang & Hengjian Cui

Authors
  1. Rui Huang
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  2. Hengjian Cui
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Corresponding author

Correspondence to Rui Huang.

Additional information

This research was supported by the Natural Science Foundation of China under Grant Nos. 11071022, 11028103, 11231010, 11471223, BCMIIS and the Beijing Municipal Educational Commission Foundation under Grant Nos. KZ201410028030, KM201210028005, and Jishou University Subject in 2014 (No: 14JD035).

This paper was recommended for publication by Editor SUN Liuquan.

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Huang, R., Cui, H. Consistency of chi-squared test with varying number of classes. J Syst Sci Complex 28, 439–450 (2015). https://doi.org/10.1007/s11424-015-3051-2

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  • DOI: https://doi.org/10.1007/s11424-015-3051-2

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