Degree | Type | Year | Semester |
---|---|---|---|
2503740 Computational Mathematics and Data Analytics | OB | 2 | 1 |
A good knowledge of the contents of the subjects studied during the first course is considered very important, especially those of Probability and Calculus.
This is the first course in the Bachelor's degree that focuses on Statistical Inference, a branch of statistics that uses data from a "representative" sample to acquire information about a population. The course is required throughout the Bachelor's degree, as it covers different concepts and techniques that serve as the basis for many of the topics introduced in upcoming courses within the Bachelor. In particular, the course will start with a brief introduction to statistics, followed by a chapter on parameter estimation (both point and based on confidence intervals), and finally chapters on frequentist-based significance tests and an introduction to classical linear regression models.
To protect everyone's safety, in-person teaching and evaluable activities will be adjusted in accordance with health authority recommendations.
Preliminaries of Probability (reminder): Probability and random variables. Law concept. Discrete-valued distributions. Density and probability functions. Expectation and variance. Moment generating function. Examples.
Topic 1. Introduction to Statistics.
1. Descriptive statistics and inferential statistics.
1.1. Basic concepts in inference: statistical population and sample; parameters, statistics and estimators.
1.2. Statistical models: parametric and non-parametric.
2. Most common statistics: the sample moments. The order statistics.
3. Distribution of some statistics.
3.1. From a sample of a Normal population: Fisher's theorem.
3.2. The Central Limit Theorem: asymptotic normality of sample moments and proportion.
Topic 2: Point estimation.
1. Point estimators: definition and properties.
1.1. Bias
1.2. Comparison of estimators without bias. Relative efficiency
1.3. Comparison of estimators with bias: the mean square error.
1.4. Consistency of an estimator.
1.5. Sufficient statistics.
2. Methods to obtain estimators.
2.1. Method of moments.
2.2. Method of maximum likelihood (MLE)
2.2.1. Invariance of the likelihood.
2.2.2. Score function and Fisher information.
2.2.3. Cramer-Rao inequality.
2.2.4. Properties of the MLE.
2.2.5. Delta method.
2.2.6. Numerical procedures for determining MLE.
Topic3. Estimation by confidence intervals.
1. Concept of confidence region and interval.
2. The "pivot" method for the construction of confidence intervals.
3. Confidence intervals for the parameters of a population.
3.1. For the mean of a Normal population with known and unknown deviations.
3.2. For the variance of a Normal population with known and unknown means.
3.3. Asymptotic confidence intervals: Wald, Score and LRT.
4. Confidence intervals for the parameters of two populations.
4.1. Confidence intervals with independent samples.
4.2. Confidence intervals for the difference of means of two Normal populations with paired data.
5. Bootstrap techniques.
Topic 4: Significance tests.
1. Introduction.
1.1. Type I and II errors.
1.2. Power function.
1.3. Tests consistency.
1.4. p-values.
1.5. Duality between confidence intervals and significance tests.
2. Tests for the parameters of a population.
2.1. For the mean of a Normal population with known and unknown deviations.
2.2. Asymptotic tests for the mean of a population when the sample is large.
2.3. For the variance of a Normal population.
3. Tests for the parameters of two populations.
3.1. Hypothesis tests with independent samples.
3.2. Tests of hypotheses with paired data.
Topic 5. Simple linear regression model.
1. Purpose of the model.
2. Ordinary least squares (OLS) estimators.
3. Inferencebased on the linear regression model.
4. Predictions.¡
IMPORTANT: In teaching, the gender perspective involves reviewing androcentric biases and questioning the assumptions and hidden gender stereotypes. This revision involves including the contents of the subjectthe knowledge produced by scientific women, often forgotten, seeking the recognition of their contributions,as well as that of their works in the bibliographical references. Efforts will also be made to introduce the most practical part of the subject, the analysis and comparison of statistical data by sex, commenting on the classroom causes and the social and cultural mechanisms that can sustain the observed inequalities.
The course is organized into lecture, exercise and lab sessions.
In lectures, we will introduce the concepts and techniques outlined in the course program. Given that the content is mostly based on the standard topics of an introduction to statistical inference course, the recommended bibliography can be used to follow the course. Lecture slides and related material will be available in Moodle. The exercise sessions are intended to work through and understand statistical concepts. Each exercise will be available in Moodle, as will the solutions (after they have been solved in sessions). The goal of the lab sessions is to learn how to apply the methods given in lectures using the statistical software R, as well as how to evaluate the findings. Outlines for lab sessions will be accessible in Moodle as well.
IMPORTANT: To work more comfortably with R, it is recommended to use the RStudio interface: it is free, "Open source" and works with Windows, Mac and Linux. https://www.rstudio.com/
OBSERVATION: The gender perspective in teaching goes beyond the contents of the subjects, since it also implies a revision of the teaching methodologies and of the interactions between the students and the teaching staff, both in the classroom and outside. In this sense, participatory teaching methodologies, where an egalitarian, less hierarchical environment is generated in the classroom, avoiding stereotyped examples in gender and sexist vocabulary, with the aim of developing critical reasoning and respect for the diversity and plurality of ideas, people and situations, tend to be more favorable to the integration and full participationof the students in the classroom, and therefore their effective implementation in this subject will besought.
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.
Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|
Type: Directed | |||
Practical classes | 12 | 0.48 | 1, 16, 2, 5, 6, 10, 8, 9, 12, 13, 17 |
Problems class | 18 | 0.72 | 16, 11, 3, 4, 5, 6, 10, 8, 9, 15, 14 |
Theory classes | 30 | 1.2 | 1, 16, 11, 3, 4, 5, 6, 10, 8, 9, 15, 14 |
Type: Autonomous | |||
Exams | 15 | 0.6 | 1, 16, 11, 3, 4, 5, 6, 10, 8, 9, 15, 14 |
Problems resolution | 25 | 1 | 1, 16, 11, 3, 4, 5, 6, 10, 8, 9, 15, 14 |
Workshop resolution | 20 | 0.8 | 1, 16, 11, 3, 2, 4, 5, 6, 10, 8, 9, 7, 15, 14, 12, 13, 17 |
The course evaluation will consist of an evaluation of the exercise sessions (score C), an evaluation of the lab sessions (score P) and the final exam (E1). In particular, score C weights 20%, score P 30% and the final exam weights 50%. The final grade of the course will be thus computed as follows:
G = 0.50 × E1 + 0.20 × C + 0.30 × P
Reset and / or improvement of the exam score:
If a student's final G score is more thanor equal to 5, he or she passes the course. Otherwise, or if the student wants to improve his or her score, the student can use the reset test to improve/reset E1 evaluation. The score of the reset exam will be E2. The final grade will be thus determined as follows:
FG = 0.50 × max (E1, E2) + 0.20 × C + 0.30 × P
Observation 1: Scores C and P are not recoverable
Observation 2: If a student writes either the E1 or E2 exams, it is considered that the student has been enrolled in the course and thus there will be an evaluation of such a course. Otherwise, the qualification will be Non presented, even if the student has evaluation on either C and/or P.
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
Final exam / Reassessment (E) | 0,50 | 10 | 0.4 | 1, 16, 11, 3, 4, 5, 6, 10, 8, 9, 15, 14 |
Practical exam (P) | 0,30 | 12 | 0.48 | 1, 16, 11, 3, 2, 4, 5, 6, 10, 8, 9, 15, 14, 12, 13, 17 |
Problems delivery (C) | 0,20 | 8 | 0.32 | 1, 16, 11, 3, 4, 5, 6, 10, 8, 9, 7, 15, 14 |
Berger, R.L., Casella, G.: Statistical Inference. Duxury Advanced Series. 2002.
Daalgard, P.: Introductory Statistics with R. Springer. 2008.
Daniel, W.W.: Biostatistics. Wiley. 1974.
DeGroot, M. H.: Schervish, M.J. Probability and Statistics. Pearson Academic. 2010.
Peña, D.: Estadística. Fundamentos de estadística. Alianza Universidad. 2001.
R Tutorial. An introduction to Statistics. https://cran.r-project.org/manuals.html. juny 2019.
Silvey, S.D.: Statistical Inference. Chapman&Hall. 1975.
Held, Sabanes Bove (2013): Applied Statistical Inference: Likelihood and Bayes.
Springer
Pawitan (2013): In all Likelihood: Statistical Modelling and Inference using
Likelihood. Oxford University Press
Young, Smith (2005): Essentials of Statistical Inference. Cambridge University Press
Cox, D.R. and Hinkley, D.V. (1979). Theoretical Statistics. 1st Edition, Chapman
and Hall/CRC