Abstract
We study the χ 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 χ 2 statistic in the case of multinomial trials. We obtain explicit expressions and estimates for the expectation and the variance of the χ 2 statistic. For the χ 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 χ 2 distribution). Conditions for the convergence to the corresponding limit distributions are given.
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Original Russian Text © B. I. Selivanov, 2008, published in Matematicheskie Zametki, 2008, Vol. 83, No. 6, pp. 899–911.
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Selivanov, B.I. Limit distributions of the χ 2 statistic of K. Pearson in a sequence of independent trials. Math Notes 83, 821–832 (2008). https://doi.org/10.1134/S0001434608050271
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DOI: https://doi.org/10.1134/S0001434608050271