# 28202 - Statistics (A. C.)

### Course Unit Page

• Teacher Gabriele Soffritti

• Credits 10

• SSD SECS-S/01

• Language Italian

• Campus of Bologna

• Degree Programme Second cycle degree programme (LM) in Statistics, Economics and Business (cod. 8876)

• Course Timetable from Nov 10, 2020 to Mar 19, 2021

### SDGs

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

## Learning outcomes

The main goal is to provide students with the theoretical and practical knowledge of the main probabilistic methods for the analysis of random variables and the parametric techniques for statistical inference based on the likelihood function.

## Course contents

First part. Probabilistic methods for the analysis of random variables

• Main probabilistic models (4 hours).
• Expected value, variance and moment-generating function of a random variable (4 hours).
• Transforming random variables (4 hours).
• Bivariate random vectors. Joint, conditional and marginal distributions. Independence. Association measures (8 hours).
• Multivariate random vectors. Joint, conditional and marginal distributions. Independence (6 hours).
• Sequences of random variables. Limit theorems and convergence (4 hours).
• Functions of a random vector (4 hours).
• The multinomial distribution. The multivariate normal distribution (4 hours).

Second part. Parametric techniques for statistical inference

• Purposes of statistical inference. Probabilistic models, sampling methods and statistical models. Sampling distributions. Identificability of a statistical model (3 hours).
• The likelihood function and the likelihood principle (2 hours).
• Statistics, sufficient statistics and minimal sufficient statistics (4 hours).
• Exponential families of distributions (3 hours).
• The problem of point estimation and the solutions obtained by using the maximum likelihood method. Other methods of estimation (outlines) (4 hours).
• Fisher information and Rao-Cramér inequality. Desirable properties of an estimator and properties of the maximum likelihood estimators (4 hours).
• The hypothesis-testing problem and the solutions obtained according to the Neyman-Pearson approach (4 hours).
• The likelihood ratio tests and its most important applications (8 hours).

The reported number of hours is an estimate which takes account of both theoretical and practical lessons. Practical lessons will take place on a weekly basis, starting from the second week of lesson; they will be given by a tutor.

Required background knowledge in mathematics and linear algebra

• Rules for product and summation notation. Factorial, binomial coefficient and their properties.
• Real functions. The main real functions and their properties.
• Limit, derivative and integration.
• Matrix algebra. Rank and inverse of a matrix. Symmetric, positive definite and orthogonal matrices. The determinant of a matrix.

Required background knowledge in probability

• Random experiments and their sample spaces. Simple, compound and disjoint events. Impossible and certain events. Events obtained by intersection, union and negation.
• Definitions and axioms of probability. Conditional probability. Independent events. The law of total probability. Bayes' theorem.
• Random variables. Rules for computing probabilities for any random variable. The distribution function of a random variable. Probability mass function. Probability density function.

Required background knowledge in statistics

• Empirical frequency distributions.
• Measures of location (mode, median, arithmetic mean).
• Measures of dispersion, linear correleation and regression.
• Fundamentals of parametric estimation and hypothesis testing.
• Statistical tables for the standard normal, chi-squared and Student's t distributions.

• J. H. McColl, Multivariate probability. Arnold, London, 2004, chapters 1-8.
• A. Azzalini, Inferenza statistica. Una presentazione basata sul concetto di verosimiglianza. 2° edizione, Springer-Verlag Italia, Milano, 2001, chapters 1-4.
• Teacher's lecture notes on the main univariate probabilistic models.
• Teacher's lecture notes with examples of exercises and questions from past written exams.

Teacher's lecture notes are available on the platform "Virtual learning environment" of the University of Bologna (https://virtuale.unibo.it/) for all enrolled students. In order to have access to this platform, students must use their username and password.

Students without the required background knowledge are supposed to fill their gap before preparing themselves for STATISTICA C.A. examination.

Students without the required background knowledge in probability should choose the course 84608 PROBABILITA', whose lessons take place just before those of the course STATISTICA C.A.

Recommended textbooks for building background knowledge in mathematics and matrix algebra

• C. Canuto, A. Tabacco. Analisi matematica 1: teoria ed esercizi. Quarta edizione. Springer, Milano, 2014.
• E. Schlesinger. Algebra lineare e geometria. Zanichelli, Bologna, 2011.
• G. Strang. Algebra lineare. Apogeo, Milano, 2008.

Recommended textbooks for building background knowledge in probability and statistics

• G. Cicchitelli. Statistica: principi e metodi. Seconda edizione. Pearson Italia, Milano - Torino, 2012.
• A. Montanari, P. Agati, D.G. Calo. Statistica con esercizi commentati e risolti. Casa Editrice Ambrosiana, Milano, 1998.
• D. Piccolo. Statistica per le decisioni: la conoscenza umana sostenuta dall'evidenza empirica. Il Mulino, Bologna, 2004.
• A. D'Elia, D. Piccolo. Statistica per le decisioni: test di autovalutazione. Il Mulino, Bologna, 2004.
• D. Piccolo. Statistica per le decisioni: un'introduzione alla statistica nell'ambiente R. Il Mulino, Bologna, 2010.
• D. Piccolo. Statistica. Il Mulino, Bologna, 2010.

## Teaching methods

Theoretical and practical lessons. Practical lessons are given by a tutor; they take place on a weekly basis starting from the second week.

Although attending lessons is not mandatory, it is strongly recommended.

## Assessment methods

The exam tests the qualifications of each student both on a theoretical and a practical level. It is written, lasts two hours and normally takes place in a room. It may be composed of either closed-ended questions or open-ended questions, some of which concern the theoretical aspects of probabilistic and inferential methods, while some other are mainly focused on the ability of using methods for practical problems. These latter questions require solving numerical exercises. Consulting textbooks or notes during the written exam is not allowed. A pocket calculator is necessary. The maximum mark for each question is fixed in advance and is visible by the student who takes the exam. The sum of the marks for all questions is 32. The overall final grade is given by the sum of the marks in the exercises, which is expressed on a scale of 30.

Depending on how the coronavirus (COVID-19) pandemic evolves, there would be the need for suitable modifications of these assessment methods. In that case, more notice will be given here.

Further useful information about the exams

• In order to take the exam, students are required to put their names down for the exam through Almaesami platform.
• Exams can only be taken in the official exam sessions.
• An identity card is required to be admitted to the exam.
• Students are allowed to withdraw from the exam.

## Teaching tools

Most explanations are given without the use of slides prepared by the teacher, with some exceptions. Such slides are available on the platform "Virtual learning environment" of the University of Bologna (https://virtuale.unibo.it/) for all enrolled students, where examples of written tests are available together with exercises that each student should solve at home.

Explanations provided by the teacher should be used to prepare the exam in conjunction with the ones available in the recommended textbooks.

## Office hours

See the website of Gabriele Soffritti