Mathematical Statistics

A.Y. 2021/2022
Overall hours
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 Statistics, the second part to Bayesian Mathematical Statistics.
During the lab activities, the fundamentals of simulations and data analyses will be provieded with advanced software instruments (Matlab, SAS, R or similar software).
Expected learning outcomes
Basic notions and theorems of univariate Mathematical Statistics.
The student will then be able to apply and broaden his/her knowledge of the subjects in different areas of interest, both in theoretical and applied contexts, and to perform statistical data analyses.
Course syllabus and organization

Single session

Lesson period
First semester
More specific information on the delivery modes of training activities for academic year 2021/22 will be provided over the coming months, based on the evolution of the public health situation
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 from 6cfu, and for Part 2 from 3cfu. The laboratory part relating to Part 2 will also be evaluated during the written test. The written test grade is out of thirty and for the 9 CFU 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 from 6cfu.
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. Interval estimation.
5. Hypothesis testing
5.1. Power function and UMP tests
5.2 The Neyman-Pearson Lemma
5.3. Likelihood ratio
5.4 Classical parametric tests
6. Further properties of estimators:
6.1. Sufficiency.
6.2. Completeness.
6.3. Methods for variance reduction: The Rao-Blackwell and Lehmann-Scheffe' Theorems.
6.4. The Cramer-Rao Theorem
6.5. Efficiency and Fisher's information.
7. Properties of maximum likelihood estimation
8. Homogeneous and inhomogeneous Poisson point process: properties and inference
9. Elements of non-parametric statistics
9.1 Inference on the cumulative function: Kolmogorov statistics, Glivenko-Cantelli theorem
9.2. Hypothesis testing on continuous distribution: Kolmogorov-Smirnov and Kolmogorov-Lilliefors test
9.3. Hypothesis testing on general distribution: Pearson statistics
9.4. Chi square test
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
Statistica Matematica (seconda parte)
Course syllabus
10. Elements of Bayesian statistics
10.1 A priori and a posteriori distributions
10.2 Conjugate families of distributions.
10.3 Bayesian estimators
10.4 Credible intervals and and Bayesian tests (hints)
10.5 Exchangeability and De Finetti's theorem.
11. Simulation and data analysis laboratory with R.
11.1. Simulation of Binomial and Poisson processes
11.2. Descriptive statistics: frequencies, estimators, quartiles, histogram and boxplot
11.3. Inferential statistics: estimators and confidence intervals
11.4. Parametric hypothesis tests
11.5. Hypothesis testing on the distribution and independence of samples.
Teaching methods
Frontal lessons for theory and exercises; computer room for the lab
Teaching Resources
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
Statistica Matematica (prima parte)
MAT/06 - PROBABILITY AND STATISTICS - University credits: 6
Practicals: 24 hours
Lessons: 36 hours
Statistica Matematica (seconda parte)
MAT/06 - PROBABILITY AND STATISTICS - University credits: 3
Practicals: 12 hours
Laboratories: 12 hours
Lessons: 9 hours
Educational website(s)
upon appointment via email
Via Saldini 50, second floor