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

Shuhua Chang, Deli Li and Yongcheng Qi

Journal of Applied Statistics, 2023, vol. 50, issue 5, 1078-1093

Abstract: 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.

Date: 2023
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DOI: 10.1080/02664763.2021.2017413

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