The problem of dealing with nuisance parameters is a fundamental aspect of statistical theory and methodology, particularly in likelihood-based inference. Commonly used approaches to eliminating a nuisance parameter from a statistical model include marginal and conditional inference and the use of the profile likelihood function. An alternative approach is to use an integrated likelihood, in which the nuisance parameter is eliminated from the likelihood function by integration with respect to a given weight function. Integrated likelihoods have the advantage that they are always available and, unlike the profile likelihood, they are based on averaging rather than maximization, which has been shown to be a more reliable approach in many models of interest. The primary drawback of the integrated likelihood approach is that weight function needed for its implementation must be chosen. The goal of this research is to study the use of integrated likelihoods in non-Bayesian, likelihood-based, inference. The most important aspect of this is the construction of the weight function so that the resulting integrated likelihood function is useful for non-Bayesian inference. Other topics considered in the research include development of higher-order asymptotic theory, development of computational algorithms, comparisons with existing methods, applications to models with a high-dimensional nuisance parameter, and the application of the methodology to models used in practice.
This research develops a new approach to statistical theory and methodology, based on the use of an integrated likelihood function. These methods are used in the analysis of virtually all statistical models and in all fields of application. In particular, integrated likelihood methods have been used in applications ranging from the reliability of computer software to the analysis of genetic data. In contrast to some other recently-developed methods, which require considerable background in advanced statistical theory, the integrated likelihood approach is relatively straightforward to understand and to implement. Thus, the results of this proposed research are useful for researchers in a wide range of fields. The results also further our understanding of the properties of statistical models and, hence, play an important role in the education of researchers in statistics and related fields.
