Chapter 1 Introduction

This is a lecture notes of Topics in Mathematical Statistics, 2021 Fall and 2022 Spring in NCCU, lectured by Professor Tzee-Ming Huang. The textbook we used is Mathematical Statistics by Jun Shao.

The goal of this lecture note is to:

  1. Review what I have learned in mathematical statistics.
  2. Teach myself something new (at least to me) in the book.
  3. Help myself (or perhaps others) understand statistics more rigorously.

First, we will introduce measure-based probability which may take readers a lot of time if they are not familiar with concepts in real analysis or measure theory. Next, we will discuss statistical decision theory that unify statistical inference such as point estimators and hypothesis tests. Last but not least we will go through some estimation techniques in parametric and nonparametric models. A point of this class is to teach us how to prove efficiency and asymptotic normality rigorously so readers may find these two chapters more detailed. Obviously, there are lots of things left in the book but I think what we have known is enough to comprehend the remains. However, I will learn and add some topics that not appeared in class if it is adequate.