Topics in high-dimensional inference

OF THE DISSERTATION Topics in High-dimensional Inference by Wenhua Jiang Dissertation Director: Professor Cun-Hui Zhang This thesis concerns three connected problems in high-dimensional inference: compound estimation of normal means, nonparametric regression and penalization method for variable selection. In the first part of the thesis, we propose a general maximum likelihood empirical Bayes (GMLEB) method for the compound estimation of normal means. We prove that under mild moment conditions on the unknown means, the GMLEB enjoys the adaptive ration optimality and adaptive minimaxity. Simulation experiments demonstrate that the GMLEB outperforms the James-Stein and several state-of-the-art threshold estimators in a wide range of settings. In the second part, we explore the GMLEB wavelet method for nonparametric regression. We show that the estimator is adaptive minimax in all Besov balls. Simulation experiments on the standard test functions demonstrate that the GMLEB outperforms several threshold estimators with moderate and large samples. Applications to high-throughput screening (HTS) data are used to show the excellent performance of the approach. In the third part, we develop a generalized penalized linear unbiased selection (GPLUS) algorithm to compute the solution paths of concave-penalized negative