{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,28]],"date-time":"2026-09-28T20:41:54Z","timestamp":1790628114723,"version":"4.1.0"},"reference-count":16,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2019,4,22]],"date-time":"2019-04-22T00:00:00Z","timestamp":1555891200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/501100008982","name":"National Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100008982","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,9,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>We establish an approximation theory for Pearson\u2019s 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\u2019s 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\u2019s alpha-particle data.<\/jats:p>","DOI":"10.1093\/biomet\/asz020","type":"journal-article","created":{"date-parts":[[2019,3,16]],"date-time":"2019-03-16T08:12:05Z","timestamp":1552723925000},"page":"716-723","source":"Crossref","is-referenced-by-count":10,"title":["Pearson\u2019s chi-squared statistics: approximation theory and beyond"],"prefix":"10.1093","volume":"106","author":[{"given":"Mengyu","family":"Xu","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Central Florida, 4000 Central Florida Blvd, Orlando, Florida 32816, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Danna","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of California, 9500 Gilman Drive, San Diego, California 92093, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wei Biao","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Chicago, 5747 S. Ellis Avenue, Illinois 60637, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2019,4,22]]},"reference":[{"key":"2019091011535858600_B1","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1214\/ss\/1177011454","article-title":"A survey of exact inference for contingency tables","volume":"7","author":"Agresti,","year":"1992","journal-title":"Statist. Sci."},{"key":"2019091011535858600_B2","first-page":"311","article-title":"Effect of high dimension: By an example of a two sample problem","volume":"6","author":"Bai,","year":"1996","journal-title":"Statist. Sinica"},{"key":"2019091011535858600_B3","first-page":"1","article-title":"Scaling unobservable constructs in social science","volume":"47","author":"Bartholomew,","year":"1998","journal-title":"Appl. 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Complex."},{"key":"2019091011535858600_B12","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1080\/01621459.1986.10478294","article-title":"Goodness-of-fit tests for log-linear models in sparse contingency tables","volume":"81","author":"Koehler,","year":"1986","journal-title":"J. Am. Statist. Assoc."},{"key":"2019091011535858600_B13","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1080\/01621459.1980.10477473","article-title":"An empirical investigation of goodness-of-fit statistics for sparse multinomials","volume":"75","author":"Koehler,","year":"1980","journal-title":"J. Am. Statist. Assoc."},{"key":"2019091011535858600_B14","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1214\/aos\/1176343006","article-title":"Central limit theorems for multinomial sums","volume":"3","author":"Morris,","year":"1975","journal-title":"Ann. 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Sci."}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/academic.oup.com\/biomet\/advance-article-pdf\/doi\/10.1093\/biomet\/asz020\/28497752\/asz020.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/academic.oup.com\/biomet\/article-pdf\/106\/3\/716\/29133234\/asz020.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,21]],"date-time":"2019-09-21T06:44:33Z","timestamp":1569048273000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/106\/3\/716\/5476362"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,22]]},"references-count":16,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,4,22]]},"published-print":{"date-parts":[[2019,9,1]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asz020","relation":{},"ISSN":["0006-3444","1464-3510"],"issn-type":[{"value":"0006-3444","type":"print"},{"value":"1464-3510","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2019,9]]},"published":{"date-parts":[[2019,4,22]]}}}