Challenges in empirical likelihood and finite mixture modelling

The parametric likelihood method has been a popular platform for inference. When the model is regular, the maximum likelihood estimator possesses many nice properties. However, when the model is not accurate or regular, these properties may no longer hold. Because of this, researchers turn to non-parametric modelling, which imposes weaker model assumptions. Empirical likelihood is a popular non-parametric platform for inference. The resulting profile empirical likelihood function has many similar properties to its parametric counterpart. The resulting empirical likelihood confidence regions are data driven, range respecting and transformation invariant. The empirical likelihood confidence regions tend to have coverage probability below the nominal level. To address this issue, researchers have proposed various methods to improve the coverage precision. However, these methods need to estimate a theoretical tuning parameter, which is difficult to estimate robustly. In my thesis, we develop a computer experiment data-driven approach to improve the coverage precision 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 that the local maximum at hand is close to the unknown true parameter value. To overcome this obstacle, we first identify 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 consistency 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 no longer enjoys these nice properties. In fact, the MLE is not well-defined because of the unboundedness of likelihood. To address this issue, researchers have proposed penalized likelihood and constrained MLE methods for consistently estimating the mixing distribution. However, the consistency of these methods is established under the assumption that the subpopulations are non-degenerate. We relax this restriction to demonstrate that the penalized MLE remains consistent even when some subpopulations degenerate. We also develop a hypothesis testing procedure for detecting degenerate subpopulations in finite Gaussian mixture models.