A Note on Karl Pearson’s 1900 Chi-Squared Test: Two Derivations of the Asymptotic Distribution, and Uses in Goodness of Fit and Contingency Tests of Independence, and a Comparison with the Exact Sample Variance Chi-Square Result

29 Pages Posted: 8 Dec 2018

See all articles by Timothy Falcon Crack

Timothy Falcon Crack

University of Otago - Department of Accountancy and Finance

Date Written: November 14, 2018

Abstract

Karl Pearson’s chi-squared test is widely known and used, both as a goodness-of-fit test for hypothesized distributions or frequencies, and in tests of independence in contingency tables. The test was introduced in Pearson (1900), but the derivation in that paper is almost incomprehensible. Two derivations of the asymptotic distribution are given here. The first uses joint characteristic functions, and the second uses a multivariate central limit theorem. Goodness-of-fit tests and contingency table tests of independence are discussed, and the asymptotic chi-square distribution result for Pearson’s test statistic is compared and contrasted with the exact chi-square result for the sample variance estimator.

Keywords: Pearson Chi-Squared Test, Asymptotic Distribution, Joint Characteristic Function, Multivariate Central Limit Theorem

JEL Classification: C12

Suggested Citation

Crack, Timothy Falcon, A Note on Karl Pearson’s 1900 Chi-Squared Test: Two Derivations of the Asymptotic Distribution, and Uses in Goodness of Fit and Contingency Tests of Independence, and a Comparison with the Exact Sample Variance Chi-Square Result (November 14, 2018). Available at SSRN: https://ssrn.com/abstract=3284255 or http://dx.doi.org/10.2139/ssrn.3284255

Timothy Falcon Crack (Contact Author)

University of Otago - Department of Accountancy and Finance ( email )

Dunedin
New Zealand

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