Title: Topics in high-dimensional and functional inference
Advisors: Professor Xuming He, Professor Vijay Nair
Committee Member: Assistant Professor XuanLong Nguyen
Abstract: High-dimensional data have taken center stage in current statistical research with the increased ability to collect and store massive amount of data. In addition, much of these data have complex structure in the form of functions, images, and so on. This presentation describes my research on two aspects of this problem. The first is on variable selection methods that are computationally efficient and have theoretical properties under high dimensional settings. The method we propose is based on a hierarchical model that places prior distributions on the regression coefficients, similar to the well-known spike and slab priors. A distinct feature of our model is that the prior distributions are sample-size dependent through which appropriate shrinkage of the coefficients is achieved. We establish strong model consistency of the proposed method, i.e., the probability of selecting the true model goes to one even when the number of covariates is exponentially large relative to the sample size. Computations are straightforward as the posterior sampling requires Gibbs sampling involving only standard distributions. The second problem develops new notions of functional depth for simultaneous inference with functional data. Specifically, the depth metrics are used to identify 100(1 - a) percent of the “deepest” sample functions, which are then used to construct simultaneous confidence bands for the underlying function of interest. We propose two notions of depth based on ranking the sample functions and study the associated properties, including asymptotic coverage. Applications of the techniques to parametric polynomial regression as well as infinite-dimensional problems will be described.