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Abstract: Empirical likelihood is a popular non-parametric method for inference. The resulting profile empirical likelihood function has many similar properties to its parametric counterpart. The empirical likelihood confidence intervals achieve higher coverage precision compared to its parametric counterpart but tend to have below nominal-level coverage probability. To address this issue, researchers have proposed adjusted empirical likelihood and Bartlett corrected empirical likelihood methods to achieve high-order coverage precision. Still, when the sample size is small, the coverage remains unsatisfactory. In my thesis, we develop a computer experiment data-driven approach to improve the coverage precisions of empirical likelihood confidence regions.
The maximum empirical likelihood estimator, just like its parametric counterpart, shares many nice properties. However, the optimal properties cannot be utilized unless we know the local maximum at hand is close to the unknown true parameter value. To overcome this obstacle, we first propose a set of conditions under which the global maximum is consistent. We then develop a global maximum test to ascertain if the local maximum at hand is, in fact, the global maximum. Furthermore, we invent a global maximum remedy to ensure global consistency by expanding the set of estimating functions under empirical likelihood.
For non-regular models such as finite normal mixture models, the MLE is not well-defined because of the unboundedness of likelihood. To address this issue, researchers have proposed penalized likelihood and constrained MLE to consistently estimate the mixing distribution. However, the consistency of these method is established under the assumption that the component covariance matrices of the true mixing distribution are non-singular. We relax this restriction to show that the penalized MLE is still consistent when component covariance matrices of the true mixing distribution are singular. We also invent a test for degeneracy of finite normal mixture model.