Convergence rate of multinomial goodness-of-fit statistics to chi-square distribution

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Abstract

Let Y = (Y1, Y2,..., Yk)' be a random vector with multinomial distribution. In this paper we investigate the convergence rate of so-called power divergence family of statistics {Iλ, λ ∈ R} introduced by Cressie and Read (1984) to chi-square distribution. It is proved that for every k ≥ 4 Pr(2nI λ (Y) < c) = Gk-1 (c) + O(n-1+μ(k-1)), where Gr (c) is the distribution function of chi-square random variable with r degrees of freedom, μ(r) = 6/(7r + 4) for 3 ≤ r ≤ 7, μ(r) = 5/(6r + 2) for r ≥ 8. This refines Zubov and Ulyanov's result (2008). The proof uses Krä tzel-Nowak's theorem (1991) on the number of integer points in a convex body with smooth boundary.

Original languageEnglish
Pages (from-to)115-131
Number of pages17
JournalHiroshima Mathematical Journal
Volume40
Issue number1
DOIs
Publication statusPublished - Mar 2010

Keywords

  • Approximation
  • Chi-square distribution
  • Krätzel-Nowak theorem
  • Powerdivergence statistics

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

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