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**Empirical likelihood is Bartlett-correctable.**
*(English)*
Zbl 0725.62042

Summary: It is shown that, in a very general setting, the empirical likelihood method for constructing confidence intervals is Bartlett-correctable. This means that a simple adjustment for the expected value of log- likelihood ratio reduces coverage error to an extremely low \(O(n^{- 2})\), where n denotes sample size. That fact makes empirical likelihood competitive with methods such as the bootstrap which are not Bartlett- correctable and which usually have coverage error of size \(n^{-1}.\)

Most importantly, our work demonstrates a strong link between empirical likelihood and parametric likelihood, since the Bartlett correction had previously only been available for parametric likelihood. A general formula is given for the Bartlett correction, valid in a very wide range of problems, including estimation of mean, variance, covariance, correlation, skewness, kurtosis, mean ratio, mean difference, variance ratio, etc. The efficacy of the correction is demonstrated in a simulation study for the case of the mean.

Most importantly, our work demonstrates a strong link between empirical likelihood and parametric likelihood, since the Bartlett correction had previously only been available for parametric likelihood. A general formula is given for the Bartlett correction, valid in a very wide range of problems, including estimation of mean, variance, covariance, correlation, skewness, kurtosis, mean ratio, mean difference, variance ratio, etc. The efficacy of the correction is demonstrated in a simulation study for the case of the mean.

### MSC:

62G15 | Nonparametric tolerance and confidence regions |

62A01 | Foundations and philosophical topics in statistics |

62G05 | Nonparametric estimation |