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)
Pages (from-to)1078-1093
Number of pages16
JournalJournal of Applied Statistics
Issue number5
StatePublished - 2023

Bibliographical note

Funding Information:
The research of Shuhua Chang was supported in part by the National Basic Research Program of China (973 Program) [grant number 2012CB955804], the National Basic Research Program [grant number 2012CB955804], the National Natural Science Foundation of China [grant number 11771322], and the Major Project of Tianjin University of Finance and Economics [grant number ZD 1302]. The research of Deli Li was partially supported by a grant from the Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada [grant number RGPIN-2019-06065]. The research of Yongcheng Qi was supported in part by the National Science Foundation [grant number DMS-1916014]. The authors would like to thank the associate editor and three referees for their constructive suggestions that have led to improvement in the paper.

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


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

PubMed: MeSH publication types

  • Journal Article


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