Pearson's chi-squared statistics: approximation theory and beyond

• 成果类型: Article
• 署名作者: Xu, Mengyu; Zhang, Danna; Wei Biaowu
• 署名单位: State University System of Florida; University of Central Florida; University of California System; University of California San Diego; University of Chicago
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asz020
• 发表日期: 2019
• 页码: 716723
• 关键词: CENTRAL LIMIT-THEOREMS

摘要

We establish an approximation theory for Pearson's chi-squared statistics in situations where the number of cells is large, by using a high-dimensional central limit theorem for quadratic forms of random vectors. Our high-dimensional central limit theorem is proved under Lyapunov-type conditions that involve a delicate interplay between the dimension, the sample size, and the moment conditions. We propose a modified chi-squared statistic and introduce an adjusted degrees of freedom. A simulation study shows that the modified statistic outperforms Pearson's chi-squared statistic in terms of both size accuracy and power. Our procedure is applied to the construction of a goodness-of-fit test for Rutherford's alpha-particle data.