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.
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, asymptotic theory, confidence interval, bootstrap method, hypothesis test, Neyman-Pearson's lemma, linear regression, least square method, Gauss-Markov's theorem
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||✔ Practical and/or problem-solving skills|
The course consists of lectures and exercises. In the exercise, the students should solve problems and submit reports.
|Course schedule||Required learning|
|Class 1||Guidance and review of probability theory||Guidance and review of probability theory|
|Class 2||Guidance and Exercise||Outline the course and solve problems related to linear algebra and probability theory.|
|Class 3||Limit theorem and Delta method||Learn the convergence of random variables and the delta method.|
|Class 4||Unbiased estimators||Understand the concept of unbiased estimators and learn unbiased estimator of the variance.|
|Class 5||Exercise||Solve problems related to the last three lectures.|
|Class 6||Statistical estimation: Fisher information and Cramer-Rao inequality||Learn Fisher information matrix, Cramer-Rao inequality and estimation accuracy of unbiased estimators.|
|Class 7||Statistical estimation: Maximum likelihood estimator||Learn the concept of the maximum likelihood estimator, and understand its statistical properties.|
|Class 8||Exercise||Solve problems related to the last two lectures.|
|Class 9||Statistical properties of the maximum likelihood estimator||Understand Statistical properties of the maximum likelihood estimator such as asymptotic consistency and asymptotic normality.|
|Class 10||Confidence interval||Learn the concept of the confidence interval and how to construct confidence intervals for some statistical models. Understand a computer-aided method of confidence interval.|
|Class 11||Exercise||Solve problems related to the last two lectures.|
|Class 12||Statistical test: concept||Learn the concept of test, and some simple examples of tests.|
|Class 13||Statistical test: 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 the last three 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 statistical properties of least squares estimator|
|Class 18||Exercise||Solve problems related to the last two 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 the last two lectures.|
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.
Course materials are provided during class.
Reference book: 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".