Applications and Extensions of Likelihood Methods

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The proposed research considers several problems in the higher-order asymptotic theory of likelihood-based inference. Many higher-order approximations apply only to the case in which the underlying data have a continuous distribution. The proposed research considers the extension of these results to the case in which the underlying data have a lattice distribution. A second aspect of the research is the development of methods for models with a hierarchical structure. Likelihood methods are generally derived under the assumption that the likelihood function is correctly specified. Of course, in practice, the probability models used are often only an approximation to the true, but unknown, models. Hence, the proposed research considers the development of methods that are based on more limited assumptions, such as moment conditions. Statistical methods based on the likelihood function play a central role in statistical theory and methodology. Many of these methods are based on approximations which may have questionable accuracy in certain cases. The proposed research develops methods of approximation with generally higher accuracy. The result is statistical methods that offer an improvement over those currently available.