Mathematical Statistics II

My Teaching @ National Tsing Hua University

Spring 2023 Instructor

• 數理統計二 (Mathematical Statistics II) (QF 314900 / QF314901)

  • Course Description & Audience: Mathematical Statistics II, as a continuation of Mathematical Statistics I, concentrates on theoretical statistical inference, including estimation theory and hypothesis testing. Our aim is to lay a solid foundation for students to explore various areas such as quantitative finance, data science, statistical signal processing, machine learning, financial mathematics, and econometrics in the future. Many of the results will be delivered in a definition-theorem-proof manner. Hence, a certain level of mathematical maturity is expected to succeed in this course.

    • Review: Basic Probability Theory

      • Discrete and Continuous Random Variables

      • Functions of Random Variables and Expectations

      • Multivariate Probability Distributions

      • Sampling Distributions  and Central Limit Theorem 

    • Basic Statistical Inference Theory 

      • Sufficiency and Data Reduction

      • Point Estimations Theory

        • Methods of Moments

        • Methods of Maximum Likelihood

      • Properties of Point Estimators 

        • Unbiasedness 

        • Consistency (WLLN and CLT)

        • Efficiency (Cramer-Rao Bound, UMVUE, and Rao-Blackwell Theorem)

      • Confidence Intervals

      • Introduction to Hypothesis Testing Theory

      • Optimal Test Theory (Most Powerful Test and Uniform Most Powerful Test)

        • Neyman-Pearson Lemma

        • Monotone Likelihood Ratio and Karlin-Rubin Theorem

        • Likelihood Ratio Tests

    • Other Statistical Tools

      • Least Square Methods (if time permitted)

      • Nonparametric and Robust Statistics (if time permitted.)

      • Bayesian Statistics (if time permitted)

Prerequisites:  A student planning to take this course should be familiar with calculus and introductory statistics. 

Time and Place: Lectures are at T7T8T9, in 206 TSMC Building (台積館). 

Instructor's Office Hours: The instructor's office is open Monday from 12:00 to 13:00 in Room 608 of the TSMC Building (台積館). Meetings are also possible at other times by appointment.

Textbooks & References: Students will be provided with significant handout material to support the lectures at no cost.  The material is mainly drawn from the following recommended textbooks.

• G. Casella and R. L. Berger, Statistical Inference, Cengage Learning, 2001.

• R. Hogg, J. McKean, A. Craig, Introduction to Mathematical Statistics, Pearson, 2018.

• D. Wackerly, W. Mendenhall, and R. L. Scheaffer, Mathematical Statistics with Applications, Thomson Brooks/Cole, 2008.

• J. A. Rice, Mathematical Statistics and Data Analysis, Cengage Learning, 2006.

Teaching Method: Lecture.

Teaching Assistant:  TBA 

Homework: Approximately weekly. 

Grading: The grade will be based on a midterm test (30%), homework (30%), and a final (40%). The instructor may exercise discretion up to 10% in each grading category.

Course Material

• Week 01: 02/14

  • Review of Basic Probability Theory

• Week 02: 02/21

  • Review of Basic Probability Theory II

• Week 03: 02/28 

  • Peace Memorial Day. No Class.

• Week 04: 03/07

  • Random Samples & Sufficient Statistic

• Week 05: 03/14

  • Sufficient Statistic II & Point Estimation Theory (Methods of Moments)

• Week 06: 03/21

  • Point Estimation Theory II (Maximum Likelihood Estimation)

• Week 07: 03/28

  • Midterm 1

• Week 08: 04/04

  • Children's Day. No Class.

• Week 09: 04/11

  • Review of Midterm 1

  • Efficiency, Cramer-Rao Bound, and UMVUE

• Week 10: 04/18

  • Efficiency II (Rao-Blackwell Theorem). & Interval Estimation.

• Week 11: 04/25

  • Interval Estimation II. 

• Week 12: 05/02

  • Hypothesis Testing Theory I: Hypothesis, Test statistic, Type-I, II errors, power functions

• Week 13: 05/09

  • Hypothesis Testing Theory II: LRT tests.

• Week 14: 05/16

  • Hypothesis Testing Theory III. Most Powerful Tests and Neyman-Pearson Lemma

• Week 15: 05/23

  • Hypothesis Testing Theory IV.  UMP Tests

• Week 16: 05/30

  • Last day of the class.

  • Monotone Likelihood Ratio & Karlin-Rubin Theorem and p-value

  • Course Evaluation

• Final Exam (06/13)

Lecture Handouts

• Chapter 7: Sufficiency and Data Reduction

• Chapter 8: Point Estimation Theory

• Chapter 9: Interval Estimation

• Chapter 10: Hypothesis Test Theory

Probability Theory (Mathematical Statistics I) Handouts

• Chapter 1: Basic Probability Theory 

• Chapter 2: Transformation and Expectations 

• Chapter 3: Common Families of Distributions

• Chapter 4: Multiple Random Variables 

• Chapter 5: Random Samples

• Chapter 6: Limiting Behaviors 

Assignments

• Assignment 01: Review of Basic Probability Theory

• Assignment 02: Review of Basic Probability Theory II

• Assignment 03: Sufficient Statistic I

• Assignment 04: Sufficiency and Method of Moments

• Assignment 05: Maximum Likelihood Estimation (Optional)

• Assignment 06: Efficiency

• Assignment 07: Efficiency II & Interval Estimations

• Assignment 08: Interval Estimations II

• Assignment 09: Hypothesis Testing I

• Assignment 10: Study Chapter 10.

• Assignment 11: Hypothesis Testing II: LRT and MP Tests

• Assignment 12: Hypothesis Testing III: MP and UMP Tests

• Assignment 13: Hypothesis Testing IV: MLR and Karlin-Rubin Theorem (Optional, No need to submit)

• Course Evaluation

Exams

• Midterm 1. 

  • Everything up to Chapter 8.

• Final Exam.

  • Everything up to Chapter 10.