This proposal is motivated by the success of Generalized Fiducial Inference as introduced by the PIs as a generalization of Fisher's fiducial argument. As a result of the many studies conducted by the PIs on the theory and applications of generalized fiducial methods the following important conclusions can be made: (a) A unified and systematic procedure is available for developing fiducial solutions for large classes of problems; (b) The fiducial approach generally leads to very efficient inference procedures and thus they are competitive with procedures developed using other approaches; (c) Fiducial procedures are asymptotically correct in large classes of problems; (d) Many fiducial distributions can also be realized as a Bayesian posterior by an appropriate choice of a prior. However, this is not always possible, which establishes that the two approaches are not equivalent in general; (e) Both the Bayesian approach and the fiducial approach lead to useful interval inference procedures as have been established in various publications in both areas. It is clear that neither approach can claim to dominate the other; (f) Both approaches typically require MCMC simulations in regards to actual numerical computation of the required posterior or fiducial distributions. After giving due consideration to areas of statistical inference where a fiducial approach is expected to lead to new and useful results, both theoretical and practical, the PIs propose to conduct research into the following topics: (a) Extensions of Interval Data fiducial framework for Generalized Linear Mixed Models together with associated computational approaches; (b) Extension of the work of the PIs to address the model selection problem within the Generalized Fiducial Inference framework; (c) Definition and investigation of the concept of a Robustified Fiducial Distribution for a parameter and development of computational methods for calculating it from data; (d) Application of robust fiducial approaches to arrive at new robust inference methods in some standard parametric examples; (e) Development of some general computational strategies for implementation of fiducial methods for complex practical problems.
This proposal studies a new approach to statistical inference based on Fisher's fiducial argument. The implications of this work will have an immediate effect on public policy. For instance, the U.S. Food and Drug Administration (FDA) guidance document spells out analysis procedures for demonstration of equivalence of two or more drug formulations. The PIs aim to show that the fiducial approach will lead to more efficient procedures, which will result in cost and time savings, an important issue for the drug industry. In metrology, the International Bureau of Weights and Measures (BIPM) in conjunction with the International Organization for Standardization (ISO), has published a ``Guide to Expression of Uncertainty in Measurements'' (GUM) which gives the procedures to be followed by national metrological institutes such as NIST in the US, NPL in UK, and PTB in Germany. A problem that is unique to metrology is that every measurement is subject to unknown and unknowable systematic errors that are often larger than random errors. The only currently known way to quantify these unknowable systematic errors is via specification of subjective distributions for them. The GUM specifies some ad hoc methods for combining data-based estimates of standard deviations for some error components and subjective estimates of uncertainty for other error components. The PIs aim to demonstrate that the fiducial method provides a new natural approach for accomplishing this. Such results are likely to influence the metrology community in modifying and improving their current procedures.
Uncertainty quantification plays an important role for solving many scientific and engineering problems, such as predicting the movement of a passenger in a car accident. Back in the 1930's, Sir Ronald Fisher, the father of statistics, proposed the fiducial approach to uncertainty quantification. In this project the PIs thoroughly studied and substantially improved Fisher's approach, and applied their new version to a wide range of modern problems, ranging from DNA sequencing for HIV prevention, volatility estimation of high frequency trading data, analysis of item response models (potential usable for standardized testing) in psychology, to sunspot modeling in solar science. During the funding period the PIs have a total of 47 peer reviewed scientific publications. The PIs took an active role in educating and preparing the next generation statisticians to serve the needs of the society. There were 6 Ph.D. students, 2 M.S. students and 1 B.S. student involved in the project. Two of the Ph.D. students are women. The two M.S. students were advised via distance at the Royal University of Phnom Penh Cambodia, as a part of a development project through Visitor Lecturer Program co-sponsored by the International Mathematical Union and the National Research Council. To date, two of the Ph.D. students have successfully completed their studies and found employment at Carnegie Mellon University and SAS Institute respectively. For dissemination of research results, the PIs gave 48 talks at various international and national conferences and invited departmental seminars.
