RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 2008, Volume 83, Issue 6, Pages 899–911 (Mi mz4050)

Limit Distributions of the $\chi^2$ Statistic of K. Pearson in a Sequence of Independent Trials

B. I. Selivanov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We study the $\chi^2$ statistic of K. Pearson in a sequence of independent and, generally, inhomogeneous trials with a fixed number of outcomes. It is assumed that the probabilities of occurrence of outcomes of the trials satisfy certain conditions. This problem statement embraces familiar results for the $\chi^2$ statistic in the case of multinomial trials. We obtain explicit expressions and estimates for the expectation and the variance of the $\chi^2$ statistic. For the $\chi^2$ statistic centered and normalized in a suitable way, we find limit distributions (the normal one, the distribution of the sum of the squares of normal random variables and, in particular, the $\chi^2$ distribution). Conditions for the convergence to the corresponding limit distributions are given.

Keywords: $\chi^2$ statistic of K. Pearson, $\chi^2$ distribution, normal distribution, goodness-of-fit test, multinomial trials, (in)homogenous trials, asymptotically normal random variable

DOI: https://doi.org/10.4213/mzm4050

Full text: PDF file (486 kB)
References: PDF file   HTML file

English version:
Mathematical Notes, 2008, 83:6, 821–832

Bibliographic databases:

Document Type: Article
UDC: 519.24
Revised: 20.11.2006

Citation: B. I. Selivanov, “Limit Distributions of the $\chi^2$ Statistic of K. Pearson in a Sequence of Independent Trials”, Mat. Zametki, 83:6 (2008), 899–911; Math. Notes, 83:6 (2008), 821–832

Citation in format AMSBIB
\Bibitem{Sel08} \by B.~I.~Selivanov \paper Limit Distributions of the $\chi^2$ Statistic of K.~Pearson in a Sequence of Independent Trials \jour Mat. Zametki \yr 2008 \vol 83 \issue 6 \pages 899--911 \mathnet{http://mi.mathnet.ru/mz4050} \crossref{https://doi.org/10.4213/mzm4050} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2451393} \zmath{https://zbmath.org/?q=an:1152.62013} \transl \jour Math. Notes \yr 2008 \vol 83 \issue 6 \pages 821--832 \crossref{https://doi.org/10.1134/S0001434608050271} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000257399900027} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-46849088601} 

• http://mi.mathnet.ru/eng/mz4050
• https://doi.org/10.4213/mzm4050
• http://mi.mathnet.ru/eng/mz/v83/i6/p899

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. M. P. Savelov, “Predelnye raspredeleniya statistiki Pirsona dlya neodnorodnoi polinomialnoi skhemy”, Diskret. matem., 29:4 (2017), 121–129
•  Number of views: This page: 287 Full text: 55 References: 24 First page: 4