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Mathematical Statistics

Code: 106081
Credits: 6
2026/2027
Degree programme Type Course
Mathematics OP 4

Contact lecturer

Name :
Isabel Serra Mochales
Email :
isabel.serra@uab.cat

Group languages

You can consult this information at the end of the document.

Prerequisites

As this is an elective course, students are expected to have successfully completed the core undergraduate courses in probability, statistics, linear algebra, and mathematical analysis.

Objectives

The general objectives of the course in Mathematical Statistics are the following:

  1. To understand the general framework of statistical modelling in terms of conditional distributions and likelihood functions, and to master linear regression models as a basic tool for inference and prediction.
  2. To develop a solid understanding of generalized linear models as a natural extension of the linear model and their use in modelling non-Gaussian data.
  3. To understand the need for specific models for extreme phenomena and the limitations of classical models in describing distribution tails.
  4. To become familiar with the foundations of extreme value theory and its application to the modelling of maxima and threshold exceedances.
  5. To understand the incorporation of temporal structures in statistical models and the role of dependence in sequential data.
  6. To become familiar with modern statistical learning techniques for flexible modelling of parameters in complex settings.

These general objectives will enable students to acquire a solid understanding of the fundamental concepts and techniques of mathematical statistics and to apply them effectively in solving problems related to the modelling of empirical phenomena, goodness-of-fit, bootstrap methods, temporal dependence, and extreme value theory.

Learning outcomes

  1. Effectively use bibliographies and electronic resources to obtain information.
  2. Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
  3. Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
  4. Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
  5. Generate innovative and competitive proposals for research and professional activities.
  6. Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
  7. Understand abstract language and understand in-depth demonstrations of some advanced theorems of probability and statistics.

Contents

  1. Statistical modelling: from regression to generalized linear models
  2. Statistical modelling of extreme values
  3. Modelling and inference in learning models and temporal structures

Learning activities and methodology

Title Hours ECTS Learning outcomes
Problems sessions 6 0.24 1, 3, 7
Computer work 24 0.96 1, 3, 5, 6
Theoretical classes 30 1.2 1, 7
Personal work 80 3.2 1, 3, 5

The statistical models and their corresponding assumptions and properties are introduced in the theoretical sessions. Emphasis will be placed on rigor in the proofs as well as on the applicability and interpretation of the methods.

The discussion will be encouraged in the classroom and theoretical problems will be proposed to deepen the topics. Problems, and practical exercises  to be performed with free software R will be proposed.

Some sections of the course could be developed by students in the form  a written  report and presented to the classmates.

Annotation: within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.

Assessment

Continuous assessment activities

Title Weight Hours ECTS Learning outcomes
Second partial test 30% 2 0.08 1, 3, 6
Tasks delivery 20% 5 0.2 1, 3, 5
Oral exposition 20% 1 0.04 1, 2, 3, 4, 5, 6
First partial test 30% 2 0.08 1, 3, 7

CONTINUOUS ASSESSMENT

The continuous assessment scheme is as follows:

NC = 0.3 P1 + 0.3 P2 + 0.4 Lli

  • P1, P2: First and second midterm exams.
  • Lli: Grade corresponding to the submitted assignments: solving theoretical and practical problems, and/or the grade of work. These must be submitted in written and oral form.

Students who do not pass continuous assessment (i.e., if NC < 5 or P₁ or P₂ < 3) may take the resit exam, which covers the 60% corresponding to P1 + P2.

A student will be considered “not assessed” if they have not been evaluated in at least 70% of the assessment items.


SINGLE ASSESSMENT

The single assessment will consist of a comprehensive exam covering all topics addressed during the course, including a computer-based part and an oral part.


Use of Artificial Intelligence (AI)

For this course, the use of Artificial Intelligence (AI) technologies is permitted exclusively as support tools, such as:

  • bibliographic or information search,
  • text or code correction,
  • translations.

Students must clearly specify which parts were generated using AI, indicate the tools used, and include a critical reflection on how these tools influenced the process and the final outcome of the assignment.

Lack of transparency regarding the use of AI in assessed activities will be considered a form of academic dishonesty and may result in partial or total grade penalties for the activity.

Bibliography

Beirlant, J., Goegebeur, Y., Teugels, J., & Segers, J. (2004). Statistics of extremes: Theory and applications. John Wiley & Sons.

Box, G. E. P., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. (2015). Time series analysis: Forecasting and control (5th ed.). Wiley.

Coles, S. (2001). An introduction to statistical modeling of extreme values. Springer.

Davison, A. C., & Hinkley, D. V. (1997). Bootstrap methods and their application. Cambridge University Press.

de Carvalho, M., Huser, R., Naveau, P., & Reich, B. J. (Eds.). (2026). Handbook of statistics of extremes. Chapman & Hall/CRC.

Efron, B., & Hastie, T. (2016). Computer age statistical inference: Algorithms, evidence, and data science. Cambridge University Press.

Hastie, T., Tibshirani, R., & Friedman, J. (2009). The elements of statistical learning: Data mining, inference, and prediction (2nd ed.). Springer.

Letac, G. (1992). Lectures on natural exponential families and their variance functions (Monografías de Matemática, No. 50). Instituto de Matemática Pura e Aplicada (IMPA).

Longin, F. (Ed.). (2017). Extreme events in finance: A handbook of extreme value theory and its applications. John Wiley & Sons.

McCullagh, P., & Nelder, J. A. (1989). Generalized linear models (2nd ed.). Chapman & Hall.

Weiß, C. H. (2018). Integer-valued time series. Springer.

Wood, S. N. (2022). Core statistics. CRC Press.


Software

Free software as R.

Course groups and languages

The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE

Type of teaching Group Language Semester Shift
(TE) Theory 1 Catalan second semester morning-mixed
(PAUL) Classroom practices 1 Catalan second semester morning-mixed