Mathematical Statistics
Code: 106081 ECTS Credits: 6| Degree | Type | Year | Semester |
|---|---|---|---|
| 2500149 Mathematics | OT | 4 | 1 |
Contact
- Name:
- Mercè Farre Cervello
- Email:
- merce.farre@uab.cat
Use of Languages
- Principal working language:
- catalan (cat)
- Some groups entirely in English:
- No
- Some groups entirely in Catalan:
- Yes
- Some groups entirely in Spanish:
- No
Prerequisites
Objectives and Contextualisation
In this course you will learn to formalize, analyze and validate a type of statistical models that are used to explain the relationships between various variables under experimental conditions of uncertainty. In the field of mathematical statistics, confidence or prediction intervals and hypothesis tests are used to interpret the results and make decisions.
The objective is to explain the behavior of a response variable in terms of other variables related to it, called regressors, explanatory or factors, which act linearly on the response. Given a model, predictions and residuals are obtained and analyzed to detect eventual anomalies and discuss possible transformations or alternative methods. The student must be aware of the hypotheses assumed to compare several models and thus be able to select the explanatory variables that make up the best possible model. Some extensions of the linear model are also introduced, such as generalized linear models, polynomial or non-linear models, for example, since they broaden the scope of modelling. The general linear model is a theoretical framework that allows formulating analysis of variance and design of experiments techniques within the linear model.
With this course, students will be able to explore and validate the theoretical properties of the general linear model, they will know some extensions, and they will be trained to model data with free software. The importance of the most important theorems in this area, as well as their proof, will be discussed in depth.
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
- 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 abstract language and understand in-depth demonstrations of some advanced theorems of probability and statistics.
Content
Preliminaries
• The simple linear model: least squrares, maximum likelihood and other estimation methods.
• Multivariate Gaussian distributions and related laws.
The multiple linear model
• The linear model. Normal equations. Properties of the coefficients’ and variance estimators. BLUE. Goodness of fit indicators.
- Estimation of the mean response and prediction of new observations.
• Sum of squares decompositions and distributions. Hypothesis tests and confidence regions. The Cochran theorem.
• Model diagnostics. Transformations.
• Outliers and influential observations.
- The multicolinearity problem. The bias problem. Model selection criteria.
Design of experiments, anova and the general linear model
• One-way analysis of variance. Multiple comparisons.
• Analysis of the variance with several factors. Interactions.
• The design of experiments setting.
• The response surface models.
- Dummy variables in regression and the general linear model.
Certain extensions of the linear model
• Random effects models. Repeated measures models.
• Generalized linear models: binomial, Poisson, etc.
• Nonlinear regression.
Methodology
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, with the aim that students will be able to model data. Some sections of the course will be developed by students in the form of work and will be a written as a short report and presented to the classroom.
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.
Activities
| 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 |
Assessment
See the datails in the catalan version.
Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| First partial exam | 0,2 | 4 | 0.16 | 7, 4, 2 |
| Oral exposition of a report | 0,2 | 1 | 0.04 | 1, 3, 6, 5, 4, 2 |
| Second partial exam | 0,3 | 4 | 0.16 | 1, 4, 2 |
| Tasks delivery | 0,3 | 1 | 0.04 | 3, 4, 2 |
Bibliography
- Peña, D.; Regresión y Diseño de Experimentos. Alianza Editorial, 2002.
- Rao, C. B., Touttenburg, H., Shalabh, Heumann, C.; Linear Models and Generalizations. 3rd edition, Springer, 2008.
- Rawlings J. O, Pantula S. G , Dickey D. A.; Applied Regression Analysis. A Research Tool. Second Edition, Springer, 1999.
- Rencher, A.C., Schaalje, G.B.; Linear Models in Statistics. Wiley-Interscience, 2008.
- Seber, G., Lee, A.; Linear Regression Analysis. Wiley Series in Probability and Statistics, 2003.
- Hay-Jahans C.; An R Companion to Linear Statistical Models. Chapman and Hall, 2012.
- Faraway, J.; Linear Models with R. Chapman&Hall/CRC, 2005.
- Faraway, J.; Extending the linear model with R. Chapman&Hall/CRC, 2006.
- Vikneswaran; An R Companion to Experimental Design. https://cran.r-project.org/doc/contrib/Vikneswaran-ED_companion.pdf
Complementary references
- McCullagh, P., Nelder, J. A.; Generalized Linear Models. Chapman&Hall, 1989.
- Clarke, B. R.; Linear Models. The theory and applications of Analysis of Variance. Wiley Series in Probability and Statistics, 2008.
- Sen, A., Srivastava, M.; Regression Analysis. Theory, Methods and Applications. Springer, 1990.
- Carmona, F.; Modelos Lineales. Universitat de Barcelona, 2005.
- Christensen, R.; Advanced Linear Modelling. Springer, 2001.
- Christensen, R.: Log-Linear Models. Springer, 1990.
- Draper, N., Smith, H.; Applied regression Analysis. Wiley, 1998.
- Chatterjee, S. & Price, B.; Regression Analysis by Example. Wiley-Interscience, third edition, 2000.
- Scheffé, H.; The Analysis of Variance, 1999.
- Montgomery, D.C., Peck, E., Vining, G.; Introduction to Linear Regression. Wiley Series in Probability and Statistics, 2001.
Software
Free software R and Rstudio.