Integrated likelihood methods provide a promising approach to likelihood inference in which any nuisance parameters in the model are eliminated by averaging the likelihood with respect to a weight function for the nuisance parameter.Such an integrated likelihood offers a number of advantages over other approaches to likelihood inference: it is always available; it is based on averaging rather than maximization, an approach that is often more reliable; it has a certain type of finite-sample optimality; by appropriate selection of the weight function it has many of the same properties of marginal and conditional likelihood functions, when either of those is available. Integrated likelihood methods combine ideas from both Bayesian and non-Bayesian inference and, hence, provide a hybrid method with many of the benefits of both approaches. These methods represent a new way of thinking about likelihood inference in models with nuisance parameters in which the traditional approach of eliminating nuisance parameters through maximization is replaced by averaging. The research will focus on three broad areas: a study of the asymptotic properties of point estimators and the associated standard errors of maximum integrated likelihood estimators; the use of integrated likelihood methods for estimation in models with an unknown function, and the application of integrated likelihood theory and methodology to models with random effects. In each of the areas, models with a high-dimensional nuisance parameter will be of particular interest. This work will lead to better understanding of the properties of likelihood-based methods of inference as well as the development of new statistical methodology based on those results.
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 are useful in complex statistical models and these methods have been used successfully 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 computation-based and 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.
