Quantile regression has in recent years emerged successfully as a powerful supplement to the more conventional least squares regression. By modeling the conditional quantile functions, the researchers are often able to gain a much more comprehensive picture of how a response variable is associated with its covariates. The prevailing approach in quantile regression is to perform analysis of the conditional quantile functions one percentile level at a time. This approach offers great modeling flexibility at the cost of statistical efficiency. The Principle Investigator proposes to develop and study new approaches to efficient modeling of conditional quantile functions. By "borrowing strength" across neighboring quantiles and utilizing a Bayesian empirical likelihood approach, the investigator aims to advance the theory, methodology, and applications of efficient quantile regression. Efficiency gain is an important consideration of any statistical research, and the proposed modeling techniques are especially helpful in the analysis of quantiles in the data-sparse areas. The Bayesian empirical likelihood approach for quantile regression can be used in conjunction with optimal weighting for semiparametric efficiency, and with Markov chain Monte Carlo sampling for effective computation in a high dimensional parameter space.The proposed models, to be called semi-local quantile models, strike to balance bias and variance; when the models do not hold exactly, the proposed estimators follow the spirit of regularization.
Inference in data-sparse areas, including but not restricted to the analysis of high tails, is highly valuable in a wide range of scientific and social studies. The proposed research is motivated by the investigator's interdisciplinary research in climate studies and public health, and will provide researchers in statistics and other fields novel tools for better understanding and quantifying relationships between measurements. The proposed activities include new opportunities for graduate students to participate in transformative research, and will enable the investigator to continue integration of research with teaching and mentoring. The investigator pursues active academic exchanges through lecturers and collaborations, and free distribution of software, for broad dissemination of the research results.
The major activities of the PI are methodological research on quantile modeling and other approaches to statistical inference without an exact likelihood. The research shows that Bayesian empirical likelihood is a very attractive approach to estimation and inference for quantile regression. A more interesting case is made for the Bayesian methods when informative priors are used to explore commonality across quantiles. It is found that modeling multiple quantile levels together is a preferable way to carry out quantile regression analysis. A theoretical framework of informative (or shrinking) priors is developed as an appropriate asymptotic analysis tool for understanding the power of the Bayesian methods. Progresses have also been made on efficient estimation of high quantiles and on model selection with high dimensional data. The new methods developed and studied under the project provide a set of powerful tools for data analysis about conditional quantiles and about the modeling of rare events. These methods are useful in climate research, public health, and genomics. The research has been disseminated through open access computer code and through presentations of the PI at variety of platforms including conferences and seminars. The project also engaged several graduate students for research training. Three doctoral students who completed their PhD under the supervision of the PI were partially supported by this NSF award.
