30549 - MATHEMATICAL STATISTICS
Course taught in English
Go to class group/s: 27
Class-group lessons delivered on campus
Students should be comfortable with probability theory, calculus, and linear algebra. Students should also have a basic knowledge of programming in R.
The course will introduce students to the mathematical foundations of statistics. After providing a broad overview of basic tools that are commonly used in exploratory data analysis, lectures will be centered on the following main topics: point estimation, hypothesis testing, confidence regions and regression. Both frequentist and Bayesian approaches will be presented in detail. The methodological part of the course will be complemented by the discussion of real data applications.
- Descriptive statistics on univariate and bivariate samples. Sample means, variances and correlations
- Point estimation: Maximum likelihood estimators; Method of moments; Bayesian estimators; M-estimators
- Hypothesis testing: p-values and statistical significance, likelihood ratio tests, Wald Test, permutation test, Person’s chi-square test, multiple testing
- Confidence regions: definition and frequentist interpretation, pivotal quantities and construction of confidence intervals, Bayesian credibility regions
- Optimality theory: sufficient statistics, estimation theory and hypothesis testing. The exponential family of distributions.
- Regression models: simple and multiple linear regression, ANOVA, logistic regression and classification, LASSO
- Nonparametric models (e.g. nonparametric regression, density estimation, Gaussian white noise), minimax estimation rates
- Additional topics: Modern high dimensional models (e.g. Matrix completion, sparse Gaussian graphical models), Neural networks and deep learning
- Deal with statistical tools that lie at the foundations of modern Data Science applications.
- Understand the mathematical background of standard statistical methods.
- Develop a multivariable thinking that is essential to understand and model large and complex datasets.
- Develop a critical thinking: identify drawbacks and merits of both the frequentist and the Bayesian approaches to statistical inference.
- Understand the underlying concept of a selection of modern, state-of-the-art statistical techniques.
- Profitably attend courses on advanced topics in Probability and Stochastic Processes, Statistics and Machine Learning
- Identify and tailor statistical models to specific experiments, with the aim of addressing estimation and hypothesis testing problems.
- Analyze real world data with the help of the statistical software R and interpret the output critically.
- Derive mathematically rigorous underpinning for statistical methods.
- Face-to-face lectures
- Exercises (exercises, database, software etc.)
Two 90-minute long theoretical lectures and one 90-minute long practical/laboratory classes per week.
1) The exam can be passed by either taking two partial tests or by taking the general exam.
2) In order to pass the exam by means of the two partial tests one has to get at least 15/25, as an average, with equal weights, with minimum mark at least equal to 12/25.
3) In order to pass the exam by means of the general exam, one has to get at least 15/25.
4) Those who pass the mid-term exam on the 20th of October 2021, may take the second partial test either on the 18th of January, 2022 or on the 2nd of February 2022.
5) Students whose average mark of the two partial exams is no larger than 15/25 or have obtained a mark of 11/25, or below, in one of the two tests must take the general exam.
6) In the Pilot research project, which is mandatory, the student has to show that she/he is able to formulate a real world statistical problem, collect or find, pre-process, describe, and analyze the corresponding data, interpret the outcome and write a max 5 page long report. Students are encouraged to work in pairs. The Pilot Research project worth 6 points.
7) At the end of the theoretical lectures approximately 15 minutes long online quizzes will be given in Kahoot/mentimeter and the best performing students will be awarded fractions of bonus points. The total number of bonus points a student can collect during the course is capped at 2 points.
8) The final grade is the sum of the two partial exams, or the general exam, (max 25 points), the pilot research project (max 6 points) and the bonus points from the online quizzes (max 2 points). Total points greater than or equal to 31 imply honors.
Both partial exams, as well as the general exam, are written tests with exercises. They aim at assessing the students' ability to solve simple problems in Mathematical Statistics. They require the application of analytical tools and univariate and multivariate calculus techniques that have been taught during the course.
Fetsje Bijma, Marianne Jonker, Aad van der Vaart, An Introduction to Mathematical Statistics, Amsterdam University Press, 2016
Johannes Schmidt-Hieber, Lecture notes (pdf will be provided)
Suggested further reading:
Larry Wasserman, All of Statistics: A Concise Course in Statistical Inference (Springer Texts is Statistics), 2005