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UBC Theses and Dissertations

Properties of empirical and adjusted empirical likelihoods Huang, Yi

Abstract

Likelihood based statistical inferences have been advocated by generations of statisticians. As an alternative to the traditional parametric likelihood, empirical likelihood (EL) is appealing for its nonparametric setting and desirable asymptotic properties. In this thesis, we first review and investigate the asymptotic and finite-sample properties of the empirical likelihood, particularly its implication to constructing confidence regions for population mean. We then study the properties of the adjusted empirical likelihood (AEL) proposed by Chen et al. (2008). The adjusted empirical likelihood was introduced to overcome the shortcomings of the empirical likelihood when it is applied to statistical models specified through general estimating equations. The adjusted empirical likelihood preserves the first order asymptotic properties of the empirical likelihood and its numerical problem is substantially simplified. A major application of the empirical likelihood or adjusted empirical likelihood is the construction of confidence regions for the population mean. In addition, we discover that adjusted empirical likelihood, like empirical likelihood, has an important monotonicity property. One major discovery of this thesis is that the adjusted empirical likelihood ratio statistic is always smaller than the empirical likelihood ratio statistic. It implies that the AEL-based confidence regions always contain the corresponding EL-based confidence regions and hence have higher coverage probability. This result has been observed in many empirical studies, and we prove it rigorously. We also find that the original adjusted empirical likelihood as specified by Chen et al. (2008) has a bounded likelihood ratio statistic. This may result in confidence regions of infinite size, particularly when the sample size is small. We further investigate approaches to modify the adjusted empirical likelihood so that the resulting confidence regions of population mean are always bounded.

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