Mathematical Statistics
A.Y. 2024/2025
Learning objectives
The main aim of the course is to introduce the basic concepts of univariate mathematical statistics, both from a theoretical and applied point of view. Some first element also of multivariate statistics will be introduced. In particular, the first part of the course will be devoted to classical mathematical parametric statistics, the second part to classical mathematical nonparametric statistics and to parametric Bayesian statistics.
Expected learning outcomes
The student will learn the basic notions and theorems of mathematical statistics, which he/she will then be able to apply to conduct statistical investigations; he/she will be able to identify the most appropriate methods for analysing and solving a problem related to the topics of the course and correctly interpret the results in order to obtain the appropriate quantitative and qualitative answers for the data in his/her possession.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Prerequisites for admission
Basic course in Probability
Assessment methods and Criteria
The exam consists of a written test and an oral test.
- In the written test, some open-ended exercises will be assigned to verify the ability to solve statistical analysis problems, both for Part 1 (6cfu) and for Part 2 (3cfu).
The written test grade is out of thirty, and for the 9cfu part it will be given by the weighted average of the marks obtained in the two distinct parts.
- The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours).
For students who will take the full 9cfu exam, there are 2 midterm tests that replace the written test of the first or second session.
There are no midterm tests for those who only take Part 1 .
The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.
- Only students who have passed the written test of the same exam session (or the midterm tests, for the January and February sessions) can access the oral exam. During the oral exam you will be asked to illustrate some of the results of the teaching program, in order to evaluate the knowledge and understanding of the topics covered, as well as the ability to know how to apply them.
The final mark is expressed out of thirty and will be communicated immediately at the end of the oral exam.
- In the written test, some open-ended exercises will be assigned to verify the ability to solve statistical analysis problems, both for Part 1 (6cfu) and for Part 2 (3cfu).
The written test grade is out of thirty, and for the 9cfu part it will be given by the weighted average of the marks obtained in the two distinct parts.
- The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours).
For students who will take the full 9cfu exam, there are 2 midterm tests that replace the written test of the first or second session.
There are no midterm tests for those who only take Part 1 .
The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.
- Only students who have passed the written test of the same exam session (or the midterm tests, for the January and February sessions) can access the oral exam. During the oral exam you will be asked to illustrate some of the results of the teaching program, in order to evaluate the knowledge and understanding of the topics covered, as well as the ability to know how to apply them.
The final mark is expressed out of thirty and will be communicated immediately at the end of the oral exam.
Statistica Matematica (prima parte)
Course syllabus
1. Random sample and statistical models. The exponential family.
2. Properties of estimators: consistency, unbiasedness, asymptotic normality.
3. Methods of finding estimators.
4. Homogeneous Poisson process: properties and inference.
5. Interval estimation.
6. Hypothesis testing.
6.1. Power function and UMP tests.
6.2. The Neyman-Pearson Lemma.
6.3. Likelihood ratio.
6.4. Classical parametric tests.
7. Simple linear regression.
8. Further properties of estimators.
8.1. Sufficiency.
8.2. Completeness.
8.3. Methods for variance reduction: The Rao-Blackwell and Lehmann-Scheffe' Theorems.
8.4. The Cramer-Rao Theorem.
8.5. Efficiency and Fisher's information.
9. Properties of maximum likelihood estimation.
2. Properties of estimators: consistency, unbiasedness, asymptotic normality.
3. Methods of finding estimators.
4. Homogeneous Poisson process: properties and inference.
5. Interval estimation.
6. Hypothesis testing.
6.1. Power function and UMP tests.
6.2. The Neyman-Pearson Lemma.
6.3. Likelihood ratio.
6.4. Classical parametric tests.
7. Simple linear regression.
8. Further properties of estimators.
8.1. Sufficiency.
8.2. Completeness.
8.3. Methods for variance reduction: The Rao-Blackwell and Lehmann-Scheffe' Theorems.
8.4. The Cramer-Rao Theorem.
8.5. Efficiency and Fisher's information.
9. Properties of maximum likelihood estimation.
Teaching methods
Frontal lessons both for theory and exercises
Teaching Resources
1. G. Casella and R.L. Berger, Statistical Inference. Second edition (2001)
2. J. Shao, Mathematical statistics. Second edition (2003)
Lecture notes will be also provided.
2. J. Shao, Mathematical statistics. Second edition (2003)
Lecture notes will be also provided.
Statistica Matematica (seconda parte)
Course syllabus
10. Elements of non-parametric statistics.
10.1. Inference on the cumulative function: Kolmogorov statistics, Glivenko-Cantelli theorem.
10.2. Hypothesis testing on continuous distribution: Kolmogorov-Smirnov and Kolmogorov-Lilliefors test.
10.3. Hypothesis testing on general distribution: Pearson statistics.
10.4. Chi-square goodness of fit test.
10.5. Chi-square test of independence.
11. Elements of parametric Bayesian statistics.
11.1. A priori and a posteriori distributions.
11.2. Conjugate families of distributions.
11.3. Bayesian estimators.
11.4. Credible intervals and and Bayesian tests (hints).
11.5. Exchangeability and De Finetti's theorem.
10.1. Inference on the cumulative function: Kolmogorov statistics, Glivenko-Cantelli theorem.
10.2. Hypothesis testing on continuous distribution: Kolmogorov-Smirnov and Kolmogorov-Lilliefors test.
10.3. Hypothesis testing on general distribution: Pearson statistics.
10.4. Chi-square goodness of fit test.
10.5. Chi-square test of independence.
11. Elements of parametric Bayesian statistics.
11.1. A priori and a posteriori distributions.
11.2. Conjugate families of distributions.
11.3. Bayesian estimators.
11.4. Credible intervals and and Bayesian tests (hints).
11.5. Exchangeability and De Finetti's theorem.
Teaching methods
Frontal lessons for theory and exercises.
Teaching Resources
1. G. Casella and R.L. Berger, Statistical Inference. Second edition (2001)
2. J. Shao, Mathematical statistics. Second edition (2003)
3. P. Hoff. A first course in Bayesian statistical methods, Springer, New York, (2009)
4. J.M. Bernaro, A.F.M. Smith, Bayesian theory, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Ltd., Chichester (1994)
Lecture notes will be also provided.
2. J. Shao, Mathematical statistics. Second edition (2003)
3. P. Hoff. A first course in Bayesian statistical methods, Springer, New York, (2009)
4. J.M. Bernaro, A.F.M. Smith, Bayesian theory, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Ltd., Chichester (1994)
Lecture notes will be also provided.
Statistica Matematica (prima parte)
MAT/06 - PROBABILITY AND STATISTICS - University credits: 6
Practicals: 36 hours
Lessons: 27 hours
Lessons: 27 hours
Professors:
Fuhrman Marco Alessandro, Villa Elena
Statistica Matematica (seconda parte)
MAT/06 - PROBABILITY AND STATISTICS - University credits: 3
Practicals: 12 hours
Lessons: 18 hours
Lessons: 18 hours
Professors:
Fuhrman Marco Alessandro, Villa Elena
Educational website(s)
Professor(s)
Reception:
Monday, 10:30 am - 1:30 pm (upon appointment, possibly suppressed for academic duties)
Department of Mathematics, via Saldini 50, office 1017.