### 2019　Mathematical Statistics

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Academic unit or major
Undergraduate major in Mathematical and Computing Science
Instructor(s)
Kanamori Takafumi  Obuchi Tomoyuki  Kawashima Takayuki
Course component(s)
Lecture / Exercise
Day/Period(Room No.)
Tue3-4(W833)  Fri3-4(W833)  Fri7-8(W833)
Group
-
Course number
MCS.T223
Credits
3
Academic year
2019
Offered quarter
3Q
Syllabus updated
2019/9/19
Lecture notes updated
-
Language used
Japanese
Access Index ### Course description and aims

Statistics is a methodology to deduce useful knowledge from data to help 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. Finally, the analysis of variance will be considered.

### Student learning outcomes

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, asymptotics, confidence interval, bootstrap method, test, Neyman-Pearson's lemma, linear regression, least square method, information criterion, analysis of variance.

### Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills ✔ Practical and/or problem-solving skills

### Class flow

The course consists of lecture and exercise. In the lecture, the contents will be explained mainly using a black board. In the exercise, the students should solve problems and submit reports.

### Course schedule/Required learning

Course schedule Required learning
Class 1 Outline of the course and basics of linear algebra and probability theory Understand the basics of linear algebra and probability theory that are used in statistics.
Class 2 Sample distributions Understand some probability distributions required in statistics.
Class 3 Exercise Solve problems related to the last two lectures.
Class 4 Problem setup of statistical estimation Understand the problem setup of statistical estimation
Class 5 Statistical estimation: Fisher information and Cramer-Rao inequality Learn Fisher information matrix and Cramer-Rao inequality and understand the relation between these concepts and unbiased estimators.
Class 6 Exercise Solve problems related to the last two lectures.
Class 7 Statistical estimation: Maximum likelihood estimator Learn the definition of the maximum likelihood estimator, and learn the derivation of the maximum likelihood estimator on some statistical models.
Class 8 Confidence interval Understand confidence intervals and how to construct confidence intervals for some statistical models.
Class 9 Exercise Solve problems related to the last two lectures.
Class 10 Bootstrap confidence intervals Understand bootstrap confidence intervals.
Class 11 Statistical test: concept Learn the concept of test, and some simple examples of tests.
Class 12 Exercise Solve problems related to the last two lectures.
Class 13 Statistical test: Neyman-Pearson Lemma Learn Neyman-Pearson Lemma that characterizes the optimality of tests.
Class 14 test of independence, likelihood-ratio test Understand test for independence and likelihood-ratio test from the standpoint of asymptotic theory
Class 15 Exercise Solve problems related to the last two lectures.
Class 16 Linear regression and least squares methods Understand the problem setup of linear regression and least squares estimator as an application of linear algebra.
Class 17 Confidence interval and statistical test for linear regression models. Learn the statistical methods including confidence interval and statistical test for linear regression models.
Class 18 Exercise Solve problems related to the last two lectures.
Class 19 Model selection in regression problems Understand statistical methods such as information criterion, regularization and cross validations for model selection problems.
Class 20 Risk optimality Learn the concept of risk optimality as a general framework of mathematical statistics
Class 21 Exercise Solve problems related to the last two lectures.
Class 22 Summary Summarize this course.

### Textbook(s)

Tatsuya Kubokawa, "Foundations of Modern Mathematical Statistics", Kyoritsu Shuppan Co., Ltd., 2017. (in Japanese)

Unspecified.

### Assessment criteria and methods

Learning achievement is evaluated by report (50%) and the final exam (50%).

### Related courses

• MCS.T212 ： Fundamentals of Probability
• MCS.T332 ： Data Analysis

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

No prerequisites, but it is expected that the students know the basics of the probability theory as taught in the course of "Fundamentals of Probability".

### Contact information (e-mail and phone)    Notice : Please replace from "[at]" to "@"(half-width character).

Lecture: Kanamori (kanamori[at]c.titech.ac.jp)
Exercise: Kawashima (kawashima.t.ai[at]m.titech.ac.jp)

### Office hours

To be announced. 