Yin, Yiyi2023-02-162023-02-162020-12https://hdl.handle.net/11299/252542University of Minnesota Ph.D. dissertation. December 2020. Major: Statistics. Advisor: Hui Zou. 1 computer file (PDF); x, 91 pages.Machine learning topics have always been an important part of statistics and statistical learning. Driven by high demand for its applications in the industry, machine learning gained more and more attention in recent decades. Gaussian graphical model can be used to estimate the precision matrix and then construct the connection graph among a group of variables. However, existing methods may assume the strong irrepresentability condition in order to establish desirable theoretical results. Their solutions may not be symmetric and hence a post-processing step is necessary to make their estimators symmetric. We propose the simple graph builder that produces a symmetric estimator. Without assuming additional structure conditions, it can recover the true graph with high probability. The rates of convergence of simple graph builder are established. In addition, it can be computed efficiently via coordinate descent. Numerical studies demonstrate that the simple graph builder has favorably performance compared with graphical lasso and graphical adaptive lasso. In recent years, deep learning and artificial neural networks have achieved unprecedented success in a wide variety of areas. These successes definitely push forward the frontiers of machine learning. Inspired by the success of neural networks, we propose the first deep learning model for nonparametric expectile regression, and name it Expectile NN. In regression analysis, conditional expectile regression of the response variable given a set of covariates is a generalization of the classical mean regression. Linear multiple expectile regression was proposed and studied in 1987. Recently, nonparametric gradient boosting and kernel learning based multiple expectile regression methods provide the estimated function more flexibility. We adopt the deep residual learning framework and propose the neural network expectile regression model Expectile NN. It is shown in substantial numerical studies on simulated and real datasets that Expectile NN has very competitive performance compared with existing methods. The specific model architecture of Expectile NN is explicitly specified so that others can easily reproduce it. Expectiles are important risk measures in finance and risk management. It connects directly with gain-loss ratio and has both coherent and elicitable properties.enThesis or Dissertation