Proposal Number: DMS PI: Thomas A. Severini Institution: Northwestern University Project: Likelihood Methods in Statistics Abstract: This research considers several problems regarding likelihood-based statistical inference. These problems naturally fall into two areas. The first area is inference based on the likelihood ratio statistic. The research considers the development of an adjustment to the square-root likelihood ratio statistic that improves the accuracy of the usual normal approximation and is easily calculated in a wide range of models. This leads to improved methods of inference in models where the accuracy of first-order asymptotic approximations is questionable. The research also studies analogues of the likelihood ratio statistic that are based on a quasi-likelihood function. The second area of research is statistical prediction analysis. The specific problems being considered include the development of general methods of prediction based on predictive pivots and conditional inference, and the analysis and comparison of predictive likelihood methods, including the predictive density based on a prior distribution. This research is concerned with the development of accurate and useful methods for drawing conclusions and making predictions based on observational data. Many statistical methods are based on approximations; in many cases, the accuracy of these approximations is questionable. One aspect of this research is the development of more accurate and reliable approximations of this type. Statistical prediction analysis is concerned with the problem of predicting future events based on currently available data. Predictive methods have applications in a wide array of fields, including medical diagnosis, environment and global change, manufacturing, and economic forecasting. This research develops general methods of constructing pre dictions and studies the relationships among the various methods of prediction.