Statistical Inference Based on an Integrated Likelihood Function

1. Intellectual Merit 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. One drawback of integrated likelihood methods is that they are computationally intensive; however, recent advances in computational methods has made the integrated likelihood approach attractive even in very complex models. 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 goal of the proposed research is to further develop the integrated likelihood methodology and to apply those ideas to new areas of application. 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.