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.
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. "
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.
|Intercultural skills||Communication skills||✔ Specialist skills||Critical thinking skills||✔ Practical and/or problem-solving skills|
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|
|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.|
Tatsuya Kubokawa, "Foundations of Modern Mathematical Statistics", Kyoritsu Shuppan Co., Ltd., 2017. (in Japanese)
Learning achievement is evaluated by report (50%) and the final exam (50%).
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".
Lecture: Kanamori (kanamori[at]c.titech.ac.jp)
Exercise: Kawashima (kawashima.t.ai[at]m.titech.ac.jp)
To be announced.