28475 - Statistical Inference (A. C.)

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

Fundamentals of probability. Random variables. Expected value, variance and moment-generating function of a random variable. The main probabilistic models. Transforming random variables.

Random vectors. Joint, conditional and marginal distributions. Independence. Sequences of random variables. Limit theorems and convergence. Functions of a random vector. The multivariate normal distribution.

Purposes of statistical inference. Probabilistic models, sampling methods and statistical models. Sampling distributions. Identificability of a statistical model.

The likelihood function and the likelihood principle. Statistics, sufficient statistics and minimal sufficient statistics. Exponential families of distributions.

The problem of point estimation and the solutions obtained by using the maximum likelihood method. Other methods of estimation (outlines). Fisher information and Rao-Cramér inequality. Desirable properties of an estimator and properties of the maximum likelihood estimators.

The hypothesis-testing problem and the solutions obtained according to the Neyman-Pearson approach. The likelihood ratio test and its most important applications.

Readings/Bibliography

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.

Teaching methods

Theoretical and practical lessons in a lecturehall. Practical lessons are given by a tutor.

Assessment methods

The exam will test the qualifications of each student both on a theoretical and a practical level.

The exam is composed of two parts: the first is mandatory, the second is optional.

The mandatory part is written. It lasts two hours and takes place in a room. Some questions concern the theoretical aspects of probabilistic and inferential methods, other questions 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. After the written exam each student is assigned a note on a scale of 30. If the note is at least 18/30, students may ask to take the second part of the exam.

The optional part is oral and consists of an additional question concerning the the theoretical aspects of probabilistic and inferential methods. After this oral exam, students are assigned a second note, that is a score between -2 and +2. The overall note is given by the sum of the two notes.

Teaching tools

Most explanations are given by writing on the blackboard. Sometimes slides are used; they can be found on the AMS Campus website, where examples of written tests are available together with the exercises that will be solved during the practical lessons.

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

A basic knowledge of mathematical analysis, linear algebra, probability and statistical inference is required.

Office hours

See the website of Gabriele Soffritti