Modern high-dimensional complex-structured datasets are usually very noisy and heavy-tailed. The project will address two major inter-related questions arising from the challenges due to these highly noisy datasets and unstable tuning-sensitive nonparametric methods. The first question concerns the noisy nature of high-dimensional complex-structured datasets. It addresses the extent to which we can believe in the results of estimation procedures designed for regression models with very light-tailed errors, while keeping in mind that such an ideal assumption may fail miserably in practice due to numerous outliers. The second question concerns the subtle and usually unstable tuning-sensitive multidimensional nonparametric methods. It aims at understanding how far alternative tuning-free and stable statistical estimation methods with additional shape constraints can be reliable. The graduate student will be involved in developing theory and methods for multi-dimensional shape constrained models, implementing algorithms, performing numerical experiments, and validating the theoretical properties of the statistical methods.
The proposed research questions share a close common tie at the level of their underlying probabilistic structures exhibited by multiplier empirical processes, a special form of the empirical processes. Unfortunately, existing tools necessarily fail to provide sharp understandings due to complex structures within the processes. The PI will develop probabilistic tools and techniques in connection with these multiplier empirical processes. These tools and techniques will play a crucial role in understanding the phase-transitional behavior of the commonly used least squares and other related frequentist and Bayes methods in regression models with heavy tails and shape constraints. The particular problems under investigation include (a) behavior of penalized least squares estimators in nonparametric models with heavy-tailed errors, (b) behavior of Bayes procedures under quasi-Gaussian likelihood with heavy-tailed errors, (c) behavior of least squares estimators in multi-dimensional isotonic and convex shape constrained models and (d) behavior of maximum likelihood estimators in general additive models with shape constraints.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
