2025 (Current Year) 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 (W9-323(W932)) / 3-4 Fri (W9-323(W932)) / 7-8 Fri (W9-323(W932))
- Class
- -
- Course Code
- MCS.T223
- Number of credits
- 210
- Course offered
- 2025
- Offered quarter
- 3Q
- Syllabus updated
- Sep 18, 2025
- Language
- Japanese
Syllabus
Course overview and goals
Statistics is a methodology for extracting useful information from data and applying it to prediction and decision making. This course provides a standard introduction to estimation theory and testing theory in mathematical statistics. First, we explain the relationship between unbiased estimators and the Cramér–Rao inequality, as well as the principle of maximum likelihood estimation. Next, we introduce the practically important concept of confidence intervals, followed by the fundamental ideas of hypothesis testing and optimal test procedures. Finally, we explain least squares estimation, confidence intervals, hypothesis testing, and model selection in linear regression.
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 lectures, standard topics in mathematical statistics will be explained. In the exercises, students will work on problems related to the lecture content and submit report assignments. A short quiz on the LMS will be given in each lecture and exercise. Students are required to bring a PC or other device to both lectures and exercises.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Guidance, overview of statistics, and review of probability theory. |
Review probability theory. |
| Class 2 | Exercise |
Work on exercises related to the lecture. |
| Class 3 | Statistical estimation problems, unbiased estimators of the expectation and variance, unbiasedness, and consistency. |
Learn about statistical estimation problems, unbiased estimators, unbiasedness, and consistency. |
| Class 4 | Fisher information and the Cramér–Rao inequality. |
Learn about Fisher information and the Cramér–Rao inequality. |
| Class 5 | Exercise |
Work on exercises related to the lecture. |
| Class 6 | Definition and computation of the maximum likelihood estimator. |
Learn about the definition and computation of the maximum likelihood estimator, a general and widely used statistical method. |
| Class 7 | Statistical properties of the maximum likelihood estimator: consistency and asymptotic normality. |
最尤推定量の統計的性質について学ぶ．特に一致性と漸近正規性を理解する． |
| Class 8 | Exercise |
Work on exercises related to the lecture. |
| Class 9 | Problem setting of confidence intervals and methods for constructing confidence intervals. |
Learn about the problem setting of confidence intervals and methods for constructing them. |
| Class 10 | Bootstrap confidence intervals. |
Learn about bootstrap confidence intervals. |
| Class 11 | Exercise |
Work on exercises related to the lecture. |
| Class 12 | Statistical testing: concept, formulation, Fisher's significance test, and t-test using the t-distribution. |
Learn about the concept and formulation of statistical testing, Fisher's significance test, and t-test. |
| Class 13 | Neyman–Pearson hypothesis testing, power, Neyman–Pearson lemma, sample size, and most powerful tests. |
Learn about Neyman–Pearson hypothesis testing, power, Neyman–Pearson lemma, sample size, and most powerful tests. |
| Class 14 | Exercise |
Work on exercises related to the lecture. |
| Class 15 | Likelihood ratio test and Wald test. |
Learn about the likelihood ratio test and the Wald test. |
| Class 16 | Linear regression model, least squares method, and properties of the multivariate normal distribution. |
Learn about the linear regression model, the least squares method, and the properties of the multivariate normal distribution. |
| Class 17 | Statistical properties of the linear regression model. |
Learn about the statistical properties of the linear regression model. |
| Class 18 | Exercise |
Work on exercises related to the lecture. |
| Class 19 | Linear regression model: confidence intervals, bootstrap confidence intervals, and hypothesis testing. |
Learn about confidence intervals, bootstrap confidence intervals, and hypothesis testing for linear regression models. |
| Class 20 | Variable selection in linear regression. |
Learn about variable selection in linear regression. |
| Class 21 | Exercise |
Work on exercises related to the lecture. |
Study advice (preparation and review)
To enhance learning effectiveness, students are expected to prepare for and review each class (including assignments) by referring to the relevant sections of the textbooks and distributed materials, taking as a guideline the time specified in the university’s study regulations.
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 (60%): quizzes and a final exam
Exercise (40%): quizzes and reports
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."