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March 2010 Convergence Rate of Multinomial Goodness-of-fit Statistics to Chi-square Distribution
Zhenisbek Assylbekov
Hiroshima Math. J. 40(1): 115-131 (March 2010). DOI: 10.32917/hmj/1270645086

Abstract

Let $\boldsymbol{Y}=\left(Y_1, Y_2,\dots, Y_k\right)'$ be a random vector with multinomial distribution. In this paper we investigate the convergence rate of so-called power divergence family of statistics $\{I^\lambda(\boldsymbol{Y}),\lambda\in\mathbb{R}\}$ introduced by Cressie and Read (1984) to chi-square distribution. It is proved that for every $k\ge4$

$\Pr(2nI^\lambda(\boldsymbol{Y})<c)=G_{k-1}(c)+O(n^{-1+\mu(k-1)}),$

where $G_r(c)$ is the distribution function of chi-square random variable with $r$ degrees of freedom, $\mu(r)={6}/{(7r + 4)}$ for $3\le r\le 7$, $\mu(r)={5}/{(6r+2)}$ for $r\ge 8$. This refines Zubov and Ulyanov's result (2008). The proof uses Krätzel-Nowak's theorem (1991) on the number of integer points in a convex body with smooth boundary.

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Zhenisbek Assylbekov. "Convergence Rate of Multinomial Goodness-of-fit Statistics to Chi-square Distribution." Hiroshima Math. J. 40 (1) 115 - 131, March 2010. https://doi.org/10.32917/hmj/1270645086

Information

Published: March 2010
First available in Project Euclid: 7 April 2010

zbMATH: 1194.62070
MathSciNet: MR2642973
Digital Object Identifier: 10.32917/hmj/1270645086

Subjects:
Primary: 62E20, 62H10
Secondary: 52A20

Rights: Copyright © 2010 Hiroshima University, Mathematics Program

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Vol.40 • No. 1 • March 2010
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