# Mathematical Statistics

Course
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

You will find the timetables for all courses and degree programmes of Leiden University in the tool MyTimetable (login). Any teaching activities that you have sucessfully registered for in MyStudyMap will automatically be displayed in MyTimeTable. Any timetables that you add manually, will be saved and automatically displayed the next time you sign in.

MyTimetable allows you to integrate your timetable with your calendar apps such as Outlook, Google Calendar, Apple Calendar and other calendar apps on your smartphone. Any timetable changes will be automatically synced with your calendar. If you wish, you can also receive an email notification of the change. You can turn notifications on in ‘Settings’ (after login).

For more information, watch the video or go the the 'help-page' in MyTimetable. Please note: Joint Degree students Leiden/Delft have to merge their two different timetables into one. This video explains how to do this.

## 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

Additional references given in class

## Registration

Please register for the course in MyStudyMap. There are two registration periods per year: registration for the fall semester opens in July and registration for the spring semester opens in December.

Please note that it is compulsory to register your participation for every exam and retake. Not being registered for a course means that you are not allowed to participate in the final exam of the course. Not being registered for an exam means your grade will not be processed.

## Contact

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

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