Mixture Distributions Based Methods of Calibration for the Empirical Log-Likelihood Ratio |
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Authors: | Jenny Jiang Min Tsao |
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Institution: | 1. Corporate Risk Management, Sun Life Financial , Toronto, Ontario, Canada Jenny.jiang@sunlife.com;3. Department of Mathematics &4. Statistics , University of Victoria , Victoria, British Columbia, Canada |
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Abstract: | Empirical likelihood ratio confidence regions based on the chi-square calibration suffer from an undercoverage problem in that their actual coverage levels tend to be lower than the nominal levels. The finite sample distribution of the empirical log-likelihood ratio is recognized to have a mixture structure with a continuous component on 0, + ∞) and a point mass at + ∞. The undercoverage problem of the Chi-square calibration is partly due to its use of the continuous Chi-square distribution to approximate the mixture distribution of the empirical log-likelihood ratio. In this article, we propose two new methods of calibration which will take advantage of the mixture structure; we construct two new mixture distributions by using the F and chi-square distributions and use these to approximate the mixture distributions of the empirical log-likelihood ratio. The new methods of calibration are asymptotically equivalent to the chi-square calibration. But the new methods, in particular the F mixture based method, can be substantially more accurate than the chi-square calibration for small and moderately large sample sizes. The new methods are also as easy to use as the chi-square calibration. |
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Keywords: | Chi-square calibration Coverage probability Empirical likelihood ratio confidence region F distribution Finite sample distributions Mixture distributions |
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