This course gives a standard introduction to mathematical statistics. In the estimation theory, starting from linear regression, the methodologies and properties of estimators such as the maximal likelihood estimator and the Bayes estimator will be explained. Following the estimation theory, the test theory will be taught. In the test theory, several tests such as two sample test, independence test, and analysis of variance will be explained. Throughout the course, optimalities of the estimation methods and the tests will be discussed based on the statistical decision theory.
Statistics is a methodology to deduce useful knowledge from data to help decision making. Through this course, the students should learn the basics of the mathematical statistics and several methods of statistics. Finally, it is expected that the students will be able to use the statistical methods. It is also expected that by learning statistics, the students will learn knowledge that is also useful for other fields such as machine learning and signal processing.
mathematical statistics, linear regression, maximum likelihood estimator, Bayes estimator, unbiased estimator, Cramer-Rao inequality, Fisher information, asymptotics, decision theory, test, two sample test, independence test, analysis of variance, confidence interval
|✔ 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 probability theory||Understand the basics of probability theory that is a basis of statistics|
|Class 2||Simple linear regression||Learn simple linear regression, understand least squares estimator, determination coefficient and so on.|
|Class 3||Exercise||Solve problems related to the last two lectures.|
|Class 4||Multiple linear regression||Understand multiple linear regression.|
|Class 5||Properties of multiple linear regression||Learn some properties of multiple linear regression such as unbiasedness, the magnitude of variance, and so on.|
|Class 6||Exercise||Solve problems related to the last two lectures.|
|Class 7||Unbiased estimator and Cramer-Rao inequality||Learn Cramer-Rao inequality and Fisher information matrix that characterize the optimality of unbiased estimators.|
|Class 8||Sufficient statistics||Understand sufficient statistics and the optimality condition of unbiased estimators given by sufficient statistics.|
|Class 9||Exercise||Solve problems related to the last two lectures.|
|Class 10||Maximum likelihood estimator||Learn the definition of the maximum likelihood estimator, and learn to be derive the maximum likelihood estimator on some statistical models.|
|Class 11||Consistency and asymptotic normality||Learn the consistency and asymptotic normality of the maximum likelihood estimator.|
|Class 12||Exercise||Solve problems related to the last two lectures.|
|Class 13||Bayes estimator||Learn the definition of the Bayes estimator and to be able to derive the Bayes estimator on several statistical models.|
|Class 14||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 15||Exercise||Solve problems related to the last two lectures.|
|Class 16||Discriminant analysis||Learn the concept of discriminant analysis.|
|Class 17||Test||Learn the concept of test, and some simple examples of tests.|
|Class 18||Exercise||Solve problems related to the last two lectures.|
|Class 19||Neyman–Pearson lemma||Learn Neyman-Pearson Lemma that characterizes the optimality of tests.|
|Class 20||Chi-square test||Learn chi-square test and understand the likelihood test is asymptotically chi-square test.|
|Class 21||Exercise||Solve problems related to the last two lectures.|
|Class 22||Analysis of variance||Learn the concept of analysis of variance and to be able to use analysis of variance in practice.|
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".