Closer asymptotic approximations for the distributions of the power divergence goodness-of-fit statistics期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Closer asymptotic approximations for the distributions of the power divergence goodness-of-fit statistics

Authors:	Timothy R C Read

Institution:	(1) Department of Statistics, University of Wisconsin, Madison

Abstract:	Summary The members of the power divergence family of statistics $2\left( {\lambda \left( {\lambda + 1} \right)} \right)^{ - 1} \sum\limits_{j = 1}^k {X_j \left {\left( {{{X_j } \mathord{\left/ {\vphantom {{X_j } {n\pi _{0j} }}} \right. \kern-\nulldelimiterspace} {n\pi _{0j} }}} \right)^2 - 1} \right]}$ all have an asymptotically equivalent χ² distribution (Cressie and Read 1]). An asymptotic expansion for the distribution function is derived which shows that the speed of convergence to this asymptotic limit is dependent on λ. Known results for Pearson'sX ² statistic and the log-likelihood ratio statistic then appear as special cases in a continuum rather than as separate (unrelated) expansions.

Keywords:	Chi-square statistic Edgeworth expansion PearsonX ² likelihood ratio second order limit distribution
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