Mathematical Statistics
A.Y. 2025/2026
Learning objectives
The main goal of the course is to introduce both the theoretical and practical aspects of mathematical statistics. Specifically, the first part of the course will focus on parametric mathematical statistics, including both classical and Bayesian approaches, with a brief overview of supervised statistical learning. The second part will cover classical non-parametric mathematical statistics and the evaluation of estimators.
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, several open-ended exercises will be assigned to assess the ability to solve problems related to statistical analysis, both for Part 1 worth 6 CFU and Part 2 worth 3 CFU.
The grade for the written test is out of thirty, and for the 9 CFU exam, it will be based on the weighted average of the grades obtained in the two separate parts.
- The duration of the written test is proportional to the number and structure of the assigned exercises, but it will not exceed three hours.
For students taking the full 9 CFU exam, two intermediate tests are scheduled, which replace the written test for the first or second exam session.
Intermediate tests are not scheduled for those taking only Part 1 worth 6 CFU.
Results of the written tests and intermediate exams will be communicated via the SIFA platform through the UNIMIA portal.
- Only students who have passed the written test for the same exam session (or the intermediate tests, for the January and February sessions) can access the oral exam.
During the oral exam, students will be asked to explain some results from the course syllabus to assess their knowledge and understanding of the topics covered, as well as their ability to apply them.
The exam is considered passed if both the written and oral tests are successfully completed.
The grade is out of thirty and will be communicated immediately at the end of the oral exam.
- In the written test, several open-ended exercises will be assigned to assess the ability to solve problems related to statistical analysis, both for Part 1 worth 6 CFU and Part 2 worth 3 CFU.
The grade for the written test is out of thirty, and for the 9 CFU exam, it will be based on the weighted average of the grades obtained in the two separate parts.
- The duration of the written test is proportional to the number and structure of the assigned exercises, but it will not exceed three hours.
For students taking the full 9 CFU exam, two intermediate tests are scheduled, which replace the written test for the first or second exam session.
Intermediate tests are not scheduled for those taking only Part 1 worth 6 CFU.
Results of the written tests and intermediate exams will be communicated via the SIFA platform through the UNIMIA portal.
- Only students who have passed the written test for the same exam session (or the intermediate tests, for the January and February sessions) can access the oral exam.
During the oral exam, students will be asked to explain some results from the course syllabus to assess their knowledge and understanding of the topics covered, as well as their ability to apply them.
The exam is considered passed if both the written and oral tests are successfully completed.
The grade is 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 for finding estimators.
4. Homogeneous Poisson process: properties and inference.
5. Interval estimation.
6. Hypothesis testing.
6.1. Powerof a test and UMP tests.
6.2. Neyman-Pearson Lemma.
6.3. Maximum likelihood ratio.
6.4. Classical parametric tests.
7. Introduction to supervised statistical learning.
7.1. The problem of statistical learning and prediction. Regression and classification.
7.2. Parametric families and cost functions: linear regression, neural networks.
7.3. In-depth analysis of the linear model with scalar input ("simple linear regression"). Parameter estimation and related statistical procedures.
8. Elements of Bayesian parametric statistics.
8.1. Prior and posterior distributions.
8.2. Conjugate families.
8.3. Bayesian estimators.
8.4. Credibility intervals and Bayesian tests (brief overview).
8.5. Exchangeability and De Finetti's theorem.
2. Properties of estimators: consistency, unbiasedness, asymptotic normality.
3. Methods for finding estimators.
4. Homogeneous Poisson process: properties and inference.
5. Interval estimation.
6. Hypothesis testing.
6.1. Powerof a test and UMP tests.
6.2. Neyman-Pearson Lemma.
6.3. Maximum likelihood ratio.
6.4. Classical parametric tests.
7. Introduction to supervised statistical learning.
7.1. The problem of statistical learning and prediction. Regression and classification.
7.2. Parametric families and cost functions: linear regression, neural networks.
7.3. In-depth analysis of the linear model with scalar input ("simple linear regression"). Parameter estimation and related statistical procedures.
8. Elements of Bayesian parametric statistics.
8.1. Prior and posterior distributions.
8.2. Conjugate families.
8.3. Bayesian estimators.
8.4. Credibility intervals and Bayesian tests (brief overview).
8.5. Exchangeability and De Finetti's theorem.
Teaching methods
Lectures are conducted both for the theory part and for 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 also be 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 also be provided.
Statistica Matematica (seconda parte)
Course syllabus
10. Elements of non-parametric statistics.
10.1. Inference on the distribution function.
10.2. Glivenko-Cantelli theorem, Kolmogorov statistic, and related tests.
10.3. Chi-square goodness-of-fit test.
10.4. Chi-square test for independence and contingency tables.
11. Other properties of estimators.
11.1. Sufficiency.
11.2. Completeness.
11.3. Variance reduction methods: Rao-Blackwell and Lehmann-Scheffé theorems.
11.4. Cramér-Rao theorem.
11.5. Efficiency and Fisher information.
11.6. Properties of maximum likelihood estimators.
10.1. Inference on the distribution function.
10.2. Glivenko-Cantelli theorem, Kolmogorov statistic, and related tests.
10.3. Chi-square goodness-of-fit test.
10.4. Chi-square test for independence and contingency tables.
11. Other properties of estimators.
11.1. Sufficiency.
11.2. Completeness.
11.3. Variance reduction methods: Rao-Blackwell and Lehmann-Scheffé theorems.
11.4. Cramér-Rao theorem.
11.5. Efficiency and Fisher information.
11.6. Properties of maximum likelihood estimators.
Teaching methods
Lectures are conducted both for the theory part and for 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.
Modules or teaching units
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
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.