17. High Dimensional Inference

This chapter delves into the intricacies of high-dimensional statistical inference, focusing on estimation and inference problems. It begins by introducing the major goals of estimation, such as finding estimators and understanding their convergence rates. The text then shifts to inference, emphasizing the importance of uncertainty assessment, confidence intervals, and hypothesis testing. A significant portion of the chapter is dedicated to high-dimensional inference, where the parameters of interest are typically larger than the sample sizes. The chapter discusses the challenges of multiple hypothesis testing, the family-wise error rate, and the false discovery rate. It also reviews important theoretical results, such as the central limit theorem and Slutsky's theorem, which are crucial for understanding the asymptotic normality of least squares. The chapter concludes with a detailed derivation of the asymptotic normality for ordinary least squares, providing a comprehensive overview of the topic.