This version of the course guide is provisional until the period for editing the new course guides ends.

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Mathematical Statistics

Code: 106081 ECTS Credits: 6
2025/2026
Degree Type Year
Mathematics OT 4

Contact

Name:
Ana Alejandra Cabaņa Nigro
Email:
anaalejandra.cabana@uab.cat

Teaching groups languages

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


Prerequisites

The competencies in algebra, analysis and probability and statistics of the first cycle of mathematics are assumed.

Objectives and Contextualisation

The general objectives of this course in mathematical statistics, are:

1. Understanding the theoretical foundations of empirical processes and their limits.
2. Explore goodness-of-fit techniques to assess the adequacy of a statistical model to observed data.
3. Study the bootstrap method as a tool for statistical inference and estimating the distribution of an estimator.
4. Analyze extreme value theory and its application in modeling rare and extreme events.
5. Develop practical skills in implementing statistical methods related to the aforementioned topics.
6. Apply the acquired knowledge in solving real-world problems and interpreting statistical results appropriately.
7. Foster critical thinking and analytical ability to evaluate and question assumptions and findings in statistical analysis.
8. Promote effective communication of statistical concepts and obtained results through technical reports and presentations.

These general objectives will help students acquire a solid understanding of fundamental concepts and techniques in mathematical statistics and apply them effectively in problem-solving related to empirical processes, goodness of fit, bootstrap, and extreme value theory.


Competences

  • Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
  • Effectively use bibliographies and electronic resources to obtain information.
  • Generate innovative and competitive proposals for research and professional activities.
  • Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
  • Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
  • 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.
  • Understand and use mathematical language.

Learning Outcomes

  1. Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
  2. Effectively use bibliographies and electronic resources to obtain information.
  3. Generate innovative and competitive proposals for research and professional activities.
  4. Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
  5. Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
  6. 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.
  7. Understand abstract language and understand in-depth demonstrations of some advanced theorems of probability and statistics.

Content

1. Statistical models for structured data (linear models, time series, etc.) per a dades estructurades ( models lineals, sèries temporals, etc.)
2. Nonparametric statistics: empirical processes, G-o-F theory, rank tests, Bootstrap.
3. Extreme value theory.


Activities and Methodology

Title Hours ECTS Learning Outcomes
Type: Directed      
Computer work 24 0.96 1, 3, 4, 2
Problems sessions 6 0.24 7, 4, 2
Theoretical classes 30 1.2 7, 2
Type: Autonomous      
Personal work 80 3.2 3, 4, 2

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

Continous Assessment Activities

Title Weighting Hours ECTS Learning Outcomes
First partial exam 02 2 0.08 7, 4, 2
Oral exposition of a report 0,2 1 0.04 1, 3, 6, 5, 4, 2
Second partial exam 0,3 2 0.08 1, 4, 2
Tasks delivery 0,4 5 0.2 3, 4, 2

Continuous Assessment

The continuous assessment scheme is as follows:

NC = 0.3 P1 + 0.3 P2 + 0.4Lli

  • P1, P2: First and second midterms, including theory, exercises, and a practical part. Submissions (Lli) will be done in class during problem-solving sessions.

  • Lli: Grade for submitted assignments: theoretical and practical problem-solving and/or a mark for independent work, in which collateral topics or theory extensions will be developed. These must be presented both in writing and orally.

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

Any student who has not been assessed in at least 70% of the items will be considered not assessable.

Single Assessment

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

Use of Artificial Intelligence (AI)

In this course, the use of Artificial Intelligence (AI) technologies is allowed only for support tasks, such as:

  • bibliographic or information searches,

  • text or code corrections,

  • translations.

Students must clearly identify which parts were generated using AI, specify the tools used, and include a critical reflection on how these tools influenced both the process and the final result of the activity.

Lack of transparency in the use of AI in assessed activities will be considered a breach of academic integrity and may lead to partial or full grade penalties, or more serious sanctions in severe cases.


Bibliography

Nonparametric Statistics:

1. Hollander, M., & Wolfe, D. A. (1999). Nonparametric Statistical Methods. Wiley.

2. Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer.

3. Gibbons, J. D., & Chakraborti, S. (2010). Nonparametric Statistical Inference. CRC Press.


Empirical Processes:
1. "Empirical Processes: Theory and Applications" by Richard D. Pollard
2. "Weak Convergence and Empirical Processes: With Applications to Statistics" by Aad van der Vaart and Jon A. Wellner
3. "Empirical Processes in M-Estimation" by Vladimir Spokoiny

Extreme Value Theory:
1. "Extreme Value Theory: An Introduction" by Laurens de Haan and Ana Ferreira
2. "An Introduction to Statistical Modeling of Extreme Values" by Stuart Coles
3. "Extreme Value Theory: An Introduction" by F.G. Bosman, C.A.J. Klaassen, and A.J. Haan

Bootstrap:
1. "An Introduction to the Bootstrap" by Bradley Efron and Robert J. Tibshirani
2. "Bootstrap Methods and their Application" by A.C. Davison and D.V. Hinkley
3. "Bootstrap Techniques for Signal Processing" by Martin R. Cramer, Janice R. Eichenberger, and R. E. Hiorns

These books provide comprehensive coverage of their respective topics and are widely recognized as valuable resources in the field.


Software

Free software R and Rstudio and Python.


Groups and Languages

Please note that this information is provisional until 30 November 2025. You can check it through this link. To consult the language you will need to enter the CODE of the subject.

Name Group Language Semester Turn
(PAUL) Classroom practices 1 Spanish first semester morning-mixed
(TE) Theory 1 Spanish first semester morning-mixed