Abstract
AbstractIt 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.
Publisher
Cambridge University Press (CUP)
Subject
Statistics, Probability and Uncertainty,General Mathematics,Statistics and Probability
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