This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
Most of statistical inferences are based on statistical models for data, so model selection plays a fundamental role in statistical inferences. There is huge amount of literature to develop model selection theory and methodologies such as AIC, BIC, bootstrap criteria, cross-validation criteria and so on. However, model selection is still an ``unsolved'' problem in the sense that there are no magic procedures to get the best model. The goal of this proposal is to develop quasi-likelihood functions of candidate models as a very accurate and natural model selection criterion. Note that the quasi-likelihood functions here are functions of models themselves instead of parameters in the models. Motivated by the modified profile likelihoods (MPLs), the investigator treats those parameters in each candidate model as nuisance parameters, and the models themselves as the values of the ``parameter'' of interest, to develop the quasi-likelihood functions of candidate models. The selected model is then the one maximizing the quasi-likelihood of models. Some simulations have shown that the proposed MPL works very well for the selection of error probability laws in location-scale models. The MPL of models has also been obtained for composite transformation models. The investigator will then develop the quasi-likelihood function of models to select variables and error probability laws in regressions and study its theoretical properties. The investigator will also develop the quasi-likelihood of models in exponential family, study its theoretical properties justifying its good performances expected in simulations and applications, and explore to apply it to regular models. Other than these, the investigator may go further to develop the quasi-likelihood to select the number of change points in the change point problems, and to select the order of AR or ARMA time series models. The investigator will also compare the proposed quasi-likelihood function with AIC, BIC, and so on to see its advantages. The investigator may study the other model selection problems in statistics and the other disciplines and carry out some practical and important applications.
Model selection is one of the fundamental tasks of scientific inquiry. The proposed quasi-likelihood of models provides a novel, very natural, universal and extraordinarily good way to select models. This novel model selection criterion would be a significant progress in solving the ``unsolved'' model selection problems in statistics and the other disciplines. Since model selection problems exist arguably in almost every discipline, the proposed quasi-likelihood of models can be broadly used in various disciplines such as statistics, signal processing, econometrics, medicine, biology, computer sciences, communication, engineering, physics and even quantitative chemistry. The investigator will collaborate with the other disciplines to solve some of their real problems.
