Poisson and Gaussian approximations of the power divergence family of statistics
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909379518791680 |
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| author | Daly, Fraser |
| author_facet | Daly, Fraser |
| contents | Consider the family of power divergence statistics based on $n$ trials, each leading to one of $r$ possible outcomes. This includes the log-likelihood ratio and Pearson's statistic as important special cases. It is known that in certain regimes (e.g., when $r$ is of order $n^2$ and the allocation is asymptotically uniform as $n\to\infty$) the power divergence statistic converges in distribution to a linear transformation of a Poisson random variable. We establish explicit error bounds in the Kolmogorov (or uniform) metric to complement this convergence result, which may be applied for any values of $n$, $r$ and the index parameter $λ$ for which such a finite-sample bound is meaningful. We further use this Poisson approximation result to derive error bounds in Gaussian approximation of the power divergence statistics. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_17014 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Poisson and Gaussian approximations of the power divergence family of statistics Daly, Fraser Probability Statistics Theory 62E17, 60F05 Consider the family of power divergence statistics based on $n$ trials, each leading to one of $r$ possible outcomes. This includes the log-likelihood ratio and Pearson's statistic as important special cases. It is known that in certain regimes (e.g., when $r$ is of order $n^2$ and the allocation is asymptotically uniform as $n\to\infty$) the power divergence statistic converges in distribution to a linear transformation of a Poisson random variable. We establish explicit error bounds in the Kolmogorov (or uniform) metric to complement this convergence result, which may be applied for any values of $n$, $r$ and the index parameter $λ$ for which such a finite-sample bound is meaningful. We further use this Poisson approximation result to derive error bounds in Gaussian approximation of the power divergence statistics. |
| title | Poisson and Gaussian approximations of the power divergence family of statistics |
| topic | Probability Statistics Theory 62E17, 60F05 |
| url | https://arxiv.org/abs/2306.17014 |