Dimension Reduction for Non-Regular Statistical Models with Applications

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

The proposal aims to develop new statistical theory and methodology on dimension reduction for high-dimensional non-regular models which allow for discontinuity with respect to a subset of the parameters or covariates. Such models arise naturally from applications in various fields, such as statistics, biostatistics, climate, marketing research, management, economics and finance. They can capture many important features of the data structure and association between the explanatory and response variables which either low-dimensional or regular models alone cannot duplicate. This proposal focuses primarily on threshold models, an important class of non-regular models which has a wide variety of applications in statistics, biostatistics, and economics. While the literature on threshold models for low-dimensional data is comprehensive, the statistical theory and methods for threshold models applied to high-dimensional data are undeveloped due to four central challenges: (I) statistical nonregularities of the estimation, (II) increasing dimensionality, (III) unknown or incomplete distributions of response variables, (IV) computational difficulties. By introducing penalization techniques, a number of related research topics are proposed for investigation. New tools for statistical inference and computational algorithms of non-regular models applied to large and high-dimensional data, for example the brain imaging data, will be developed. These new developments will allow scientists to efficiently analyze data with substantially increased flexibility, interpretability and reduced modeling biases. In addition, the investigator will integrate new mathematical, probabilistic and computational tools with those in sciences and engineering. Dissemination of these developments will enhance new knowledge discoveries, and strengthen interdisciplinary collaborations. The research will also serve an educational purpose through multi-disciplinary courses on the contemporary state-of-the-art data mining and machine learning, and benefit the training and learning of undergraduate, graduate students and underrepresented minorities.