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On approximating some statistics of goodness-of-fit tests in the case of three-dimensional discrete data

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Abstract

We study the rate of weak convergence of the distributions of the statistics {t λ (Y), λ ∈ ℝ} from the power divergence family of statistics to the χ 2 distribution. The statistics are constructed from n observations of a random variable with three possible values. We show that

$\Pr (t_\lambda (Y) < c) = G_2 (c) + O(n^{ - 50/73} (\log n)^{315/146} ),$

where G 2(c) is the χ 2 distribution function of a random variable with two degrees of freedom. In the proof we use Huxley’s theorem of 1993 on approximating the number of integer points in a plane convex set with smooth boundary by the area of the set.

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Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    Zh. A. Asylbekov & V. V. Ulyanov

  2. Joint Stock Commercial Bank “National Clearing Center,”, Moscow, Russia

    V. N. Zubov

Authors
  1. Zh. A. Asylbekov
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  2. V. N. Zubov
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  3. V. V. Ulyanov
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Correspondence to V. V. Ulyanov.

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Original Russian Text Copyright © 2011 Asylbekov Zh. A., Zubov V. N., and Ulyanov V. V.

The authors were supported by the Russian Foundation for Basic Research (Grant 11-01-00515).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 52, No. 4, pp. 728–744, July–August, 2011.

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Asylbekov, Z.A., Zubov, V.N. & Ulyanov, V.V. On approximating some statistics of goodness-of-fit tests in the case of three-dimensional discrete data. Sib Math J 52, 571–584 (2011). https://doi.org/10.1134/S0037446611040021

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  • DOI: https://doi.org/10.1134/S0037446611040021

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