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2023　Mathematical Statistics

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Academic unit or major
Undergraduate major in Mathematical and Computing Science
Instructor(s)
Kanamori Takafumi  Kawashima Takayuki 
Class Format
Lecture / Exercise    (Face-to-face)
Media-enhanced courses
Day/Period(Room No.)
Tue3-4(W9-326(W935))  Fri3-4(W9-326(W935))  Fri7-8(W9-326(W935))  
Group
-
Course number
MCS.T223
Credits
3
Academic year
2023
Offered quarter
3Q
Syllabus updated
2023/9/12
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

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.

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, asymptotic theory, confidence interval, bootstrap method, hypothesis test, Neyman-Pearson's lemma, linear regression, least square method, Gauss-Markov's theorem

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 lectures and exercises. In the exercise, the students should solve problems and submit reports.

Course schedule/Required learning

  Course schedule Required learning
Class 1 The convergence of random variables and Slutsky's theorem. Learn the convergence of random variables and Slutsky's theorem.
Class 2 Statistical estimation Understand unbiased estimators and consistent estimators.
Class 3 Exercise 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 Maximum likelihood estimator Learn the concept of the maximum likelihood estimator, and understand its statistical properties.
Class 6 Exercise Solve problems related to lectures.
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 Confidence interval Learn the concept of the confidence interval and how to construct confidence intervals for some statistical models.
Class 9 Exercise Solve problems related to lectures.
Class 10 Bootstrap Confidence interval Understand a computer-aided bootstrap method of confidence interval.
Class 11 Statistical hypothesis testing Learn the concept of statistical test, and some simple examples of tests.
Class 12 Exercise Solve problems related to lectures.
Class 13 Neyman-Pearson Lemma Learn Neyman-Pearson Lemma that characterizes the optimality of tests.
Class 14 Likelihood-ratio test Learn likelihood-ratio test and understand its asymptotic property.
Class 15 Exercise Solve problems related to 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 Statistical properties of least squares estimator Learn Gauss-Markov theorem and understand the statistical properties of the least squares estimator
Class 18 Exercise Solve problems related to lectures.
Class 19 Confidence interval and statistical test for linear regression models. Learn confidence interval and statistical test for linear regression models.
Class 20 Summary Summarize this course.
Class 21 Exercise Solve problems related to lectures.

Out-of-Class Study Time (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, "Foundations of Modern Mathematical Statistics", Kyoritsu Shuppan Co., Ltd., 2017. (in Japanese)

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.)

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.

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