Pearson's goodness-of-fit tests for sparse distributions

Shuhua Chang, Deli Li, Yongcheng Qi

Research output: Contribution to journalArticlepeer-review


Pearson's chi-squared test is widely used to test the goodness of fit between categorical data and a given discrete distribution function. When the number of sets of the categorical data, say k, is a fixed integer, Pearson's chi-squared test statistic converges in distribution to a chi-squared distribution with k−1 degrees of freedom when the sample size n goes to infinity. In real applications, the number k often changes with n and may be even much larger than n. By using the martingale techniques, we prove that Pearson's chi-squared test statistic converges to the normal under quite general conditions. We also propose a new test statistic which is more powerful than chi-squared test statistic based on our simulation study. A real application to lottery data is provided to illustrate our methodology.

Original languageEnglish (US)
JournalJournal of Applied Statistics
StateAccepted/In press - 2022

Bibliographical note

Publisher Copyright:
© 2021 Informa UK Limited, trading as Taylor & Francis Group.


  • chi-square approximation
  • discrete distribution
  • Goodness-of-fit
  • normal approximation
  • sparse distribution


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