# Mathematical Statistics

Vak
2024-2025

The course requires tools from various areas in mathematics such as measure theory and function spaces. We briefly introduce these concepts at the beginning of the course. As we will otherwise not be able to cover interesting theory, the idea is to discuss some of these underlying concepts in less depth. An introduction to mathematical statistics, measure theory, theory of stochastic processes and functional analysis will therefore be very helpful but it is not required.

## Description

In statistics, we estimate/reconstruct objects from data. Given a noisy image for example, the aim is to reconstruct the underlying true image.
A first course in statistics typically deals with reconstruction of finite dimensional parameters, such as the mean or the standard deviation of a distribution. For many interesting applications, however, we want to assume as little as possible about the true underlying objects. Taking a fixed number of parameters is then not appropriate. Instead this should be modelled by assuming a high-dimensional or even infinite dimensional parameter space. To reconstruct an image, for instance, we can think of it as a two-dimensional function and take as a parameter space a function class.
The mathematical theory of complex statistical models has been developed largely during the past years but remains a topic of active research with many challenging open problems. Moreover, recent advances in technology and data sampled made available data sampled at increasingly high frequency and complex form. Such data are often called functional and mathematically they can be taken to be random element in Hilbert Spaces.
The course deals with the theoretical foundation of modern non parametric statistical methods. It will be concluded with an overview of modern concepts in Optimal Transport and functional data analysis.

## Course Objectives

In the course we lay some mathematical background to fundamental statistical concepts. By the end of the course, students are expected to be familiar with the concept and the manipulation of the most common loss functions, shrinkage and penalised estimators, and classical non-parametric techniques .

## Timetable

In MyTimetable, you can find all course and programme schedules, allowing you to create your personal timetable. Activities for which you have enrolled via MyStudyMap will automatically appear in your timetable.

Questions? Watch the video, read the instructions, or contact the ISSC helpdesk.

Note: Joint Degree students from Leiden/Delft need to combine information from both the Leiden and Delft MyTimetables to see a complete schedule. This video explains how to do it.

## Mode of instruction

Blackboard lectures and homework. Class notes will be made available after the lesson. We start with a short introduction of mathematical prerequisites. We then discuss general estimation methods and non parametric regression. The final third of the course will be devoted to teach the theoretical fundation of functional data analysis. To illustrate the mathematical theory we discuss applications in linguistics and image reconstruction. One lecture will be devoted to the statistical theory of neural networks.

## Assessment method

Weekly homework assignments with math problems (1/3) and a final exam (2/3).
Homework counts as practical and there is no retake for it.
Depending on the number of students the final exam will be oral or written.

L. Wasserman, All of Nonparametric Statistics

F. Ferraty, P. Vieu, Nonparametric Functional Data Analysis

## Registration

As a student, you are responsible for enrolling on time through MyStudyMap.

In this short video, you can see step-by-step how to enrol for courses in MyStudyMap.
Extensive information about the operation of MyStudyMap can be found here.

There are two enrolment periods per year:

• Enrolment for the fall opens in July

• Enrolment for the spring opens in December

Note:

• It is mandatory to enrol for all activities of a course that you are going to follow.

• Your enrolment is only complete when you submit your course planning in the ‘Ready for enrolment’ tab by clicking ‘Send’.

• Not being enrolled for an exam/resit means that you are not allowed to participate in the exam/resit.

## Contact

Docent: Valentina Masarotto, v.masarotto@math.leideuniv.nl
Please refer to the brightspace page for the TA

## Remarks

Software
Starting from the 2024/2025 academic year, the Faculty of Science will use the software distribution platform Academic Software. Through this platform, you can access the software needed for specific courses in your studies. For some software, your laptop must meet certain system requirements, which will be specified with the software. It is important to install the software before the start of the course. More information about the laptop requirements can be found on the student website.