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, the maximal likelihood estimator and the Bayes estimator will be explained. 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. "
linear regression, unbiased estimator, maximum likelihood estimator, Bayes estimator, Cramer-Rao inequality, Fisher information, asymptotics, confidence interval, test, Neyman-Pearson's lemma, analysis of variance.
|✔ Specialist skills||Intercultural skills||Communication 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||Linear regression and least squares methods||Understand the problem setup of linear regression and least squares estimator as an application of linear algebra.|
|Class 3||Exercise||Solve problems related to the last two lectures.|
|Class 4||Statistical properties of least squares methods||Learn some properties of linear regression such as unbiasedness, the magnitude of variance, and so on.|
|Class 5||Statistical estimation: problem setup||Understand the problem setup of statistical estimation|
|Class 6||Exercise||Solve problems related to the last two lectures.|
|Class 7||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 8||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 9||Exercise||Solve problems related to the last two lectures.|
|Class 10||Asymptotic theory: Consistency and asymptotic normality of the maximum likelihood estimator||Learn the consistency and asymptotic normality of the maximum likelihood estimator.|
|Class 11||Bayes estimator||Learn the definition of the Bayes estimator and to be able to derive the Bayes estimator on several statistical models.|
|Class 12||Exercise||Solve problems related to the last two lectures.|
|Class 13||Some properties of the Bayes estimator: admissibility, minimax optimality, and others||Learn some concepts of the statistical decision theory such as admissibility and minimax optimality and understand when the Bayes estimator satisfies those properties.|
|Class 14||Confidence interval||Understand the confidence interval and be able to construct confidence interval on several statistical models.|
|Class 15||Exercise||Solve problems related to the last two lectures.|
|Class 16||Statistical test: concept||Learn the concept of test, and some simple examples of tests.|
|Class 17||Statistical test: Neyman-Pearson Lemma||Learn Neyman-Pearson Lemma that characterizes the optimality of tests.|
|Class 18||Exercise||Solve problems related to the last two lectures.|
|Class 19||Confidence interval and statistical test for linear regression models.||Learn the statistical methods including confidence interval and statistical test for linear regression models.|
|Class 20||Analysis of variance||Learn the concept of analysis of variance and to be able to use analysis of variance in practice.|
|Class 21||Exercise||Solve problems related to the last two lectures.|
|Class 22||Summary||Summarize this course.|
Nobuo Inagaki "Mathematical statistics"
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".