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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.
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
Linking options:
https://www.mathnet.ru/eng/mzm4050https://doi.org/10.4213/mzm4050 https://www.mathnet.ru/eng/mzm/v83/i6/p899
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Abstract page: | 531 | Full-text PDF : | 193 | References: | 63 | First page: | 4 |
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