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

This article is cited in 1 scientific paper (total in 1 paper)

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

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

Bibliographic databases:

Document Type: Article
UDC: 519.24
Received: 23.03.2006
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
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\paper Limit Distributions of the $\chi^2$ Statistic of K.~Pearson in a Sequence of Independent Trials
\jour Mat. Zametki
\yr 2008
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\issue 6
\pages 899--911
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    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  mathnet  crossref  elib
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