2024 Faculty Courses School of Computing Undergraduate major in Mathematical and Computing Science
Mathematical Statistics
- Academic unit or major
- Undergraduate major in Mathematical and Computing Science
- Instructor(s)
- Takafumi Kanamori / Takayuki Kawashima
- Class Format
- Lecture/Exercise (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Tue / 3-4 Fri / 7-8 Fri
- Class
- -
- Course Code
- MCS.T223
- Number of credits
- 210
- Course offered
- 2024
- Offered quarter
- 3Q
- Syllabus updated
- Mar 17, 2025
- Language
- Japanese
Syllabus
Course overview and goals
Statistics is a methodology of deducing useful knowledge from data for prediction and decision making. This course gives a standard introduction to mathematical statistics. In the estimation theory, the methodologies and properties of estimators such as the linear regression estimator, the unbiased estimator and the maximal likelihood estimator will be explained. By following the estimation theory, the construction of confidential interval will be taught. In the test theory, the concept of the null and alternative hypotheses and Neyman-Pearson lemma will be introduced. The confidence interval and statistical testing for linear regression models will be explained.
Course description and aims
Objective to attain: Obtain basic knowledge about statistical methods including estimation and testing.
Theme: This course deals with the basic concepts and principles of mathematical statistics. It also enhances the development of
students’ skill in estimating the statistical structure behind observed data. "
Keywords
Unbiased estimator, maximum likelihood estimator, Cramer-Rao inequality, Fisher information, asymptotic theory, confidence interval, bootstrap method, Fisher's significance test, Neyman-Pearson's lemma, linear regression, least square method, variable selection.
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
The course consists of lectures and exercises. In the exercise, the students should solve problems and submit reports.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | The convergence of random variables and Slutsky's theorem. | Learn the convergence of random variables and Slutsky's theorem. |
| Class 2 | Exercise | Solve problems related to lectures |
| Class 3 | Unbiased estimators | Solve problems related to lectures. |
| Class 4 | Fisher information and Cramer-Rao inequality | Learn Fisher information matrix, Cramer-Rao inequality, and estimation accuracy of unbiased estimators. |
| Class 5 | Exercise | Solve problems related to lectures. |
| Class 6 | Maximum likelihood estimator | Learn the concept of the maximum likelihood estimator, and understand its statistical properties. |
| Class 7 | Statistical properties of the maximum likelihood estimator | Leaern the delta method for statistical asymptotic theory. Understand Statistical properties of the maximum likelihood estimator such as asymptotic consistency and asymptotic normality. |
| Class 8 | Exercise | Solve problems related to lectures. |
| Class 9 | Confidence interval | Learn the concept of the confidence interval and how to construct confidence intervals for some statistical models. |
| Class 10 | Bootstrap Confidence interval | Understand a computer-aided bootstrap method of confidence interval. |
| Class 11 | Exercise | Solve problems related to lectures. |
| Class 12 | Statistical hypothesis testing | Learn the concept of statistical test, and some simple examples of tests. |
| Class 13 | Neyman-Pearson Lemma | Learn Neyman-Pearson Lemma that characterizes the optimality of tests. |
| Class 14 | Exercise | Solve problems related to lectures. |
| Class 15 | Likelihood-ratio test | Learn likelihood-ratio test and understand its asymptotic property. |
| Class 16 | Exercise | Solve problems related to lectures. |
| Class 17 | Linear regression and least squares methods | Understand the problem setup of linear regression and least squares estimator as an application of linear algebra. |
| Class 18 | Confidence interval and statistical test for linear regression models. | Learn confidence interval and statistical test for linear regression models. |
| Class 19 | Exercise | Solve problems related to lectures. |
| Class 20 | Variable Selection for linear regression models. | Learn variable selection methods for linear regression models. |
| Class 21 | Summary | Summarize this course. |
| Class 22 | Exercise | Solve problems related to lectures. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend a certain length of time outside of class on preparation and review (including for assignments), as specified by the Tokyo Institute of Technology Rules on Undergraduate Learning (東京工業大学学修規程) and the Tokyo Institute of Technology Rules on Graduate Learning (東京工業大学大学院学修規程), for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
Unspecified.
Reference books, course materials, etc.
Course materials are provided during class.
Reference book: Tatsuya Kubokawa, "Introduction to Mathematical Statistics for Data Analysis," Kyoritsu Shuppan Co., Ltd., 2023. (in Japanese)
Evaluation methods and criteria
Lecture: quiz(20%), final exam(40%)
Exercise: report(40%)
Related courses
- MCS.T212 ： Fundamentals of Probability
- MCS.T332 ： Data Analysis
Prerequisites
The students are expected to know the basics of probability theory as taught in the course "Fundamentals of Probability." ▽アWatch the video of "the review of Probability Theory" in T2SCHOLA by the first lecture.