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Bounds for the chi-square approximation of the power divergence family of statistics

Part of: Distribution theory Limit theorems

Published online by Cambridge University Press:  09 August 2022

Robert E. Gaunt
Robert E. Gaunt*
Affiliation:
The University of Manchester
*
*Postal address: Department of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK. Email: [email protected]
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Abstract

It is well known that each statistic in the family of power divergence statistics, across n trials and r classifications with index parameter $\lambda\in\mathbb{R}$ (the Pearson, likelihood ratio, and Freeman–Tukey statistics correspond to $\lambda=1,0,-1/2$ , respectively), is asymptotically chi-square distributed as the sample size tends to infinity. We obtain explicit bounds on this distributional approximation, measured using smooth test functions, that hold for a given finite sample n, and all index parameters ( $\lambda>-1$ ) for which such finite-sample bounds are meaningful. We obtain bounds that are of the optimal order $n^{-1}$ . The dependence of our bounds on the index parameter $\lambda$ and the cell classification probabilities is also optimal, and the dependence on the number of cells is also respectable. Our bounds generalise, complement, and improve on recent results from the literature.

Keywords

Power divergence statisticlikelihood ratio statisticPearson’s statisticchi-square approximationrate of convergenceStein’s method

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 62E17: Approximations to distributions (nonasymptotic)
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 4 , December 2022 , pp. 1059 - 1080
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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