24018 - Statistics and Probability Theory (Advanced Course)

Learning outcomes

The main learning objectives are:

1.  to provide the student with the knowledge of

•  the axiomatic foundations of the main tools for treating a random phenomenon;
•  the theoretical foundations of the main parametric methods for statistical inference, with a special emphasis on the methods that are based on the concept of likelihood;

2. to develop his skill at using probabilistic and statistical tools to investigate a real phenomenon.

Course contents

Part I. Probability Theory (Patrizia Agati)

• Random events
• Axiomatic foundations and probability laws
• Random variables and probability distributions
• Functions of a random variable
• Moments of a random variables and moment inequalities
• Probability generating functions, moment generating functions, characteristic functions
• Some probability distributions
• Multiple random variables: joint and marginal distributions; conditional distributions and stochastic independence; covariance and correlation
• Limit theorems: convergence criteria, laws of large numbers, central limit theorem

Part II. Statistics (Gabriele Soffritti)

• Variables, statistical units, populations and samples. 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.
• The interval estimation problem and the solutions obtained according to the Neyman approach.

Readings/Bibliography

A. Azzalini, 2001 (2° edizione), Inferenza statistica. Una presentazione basata sul concetto di verosimiglianza,  Springer-Verlag Italia, Milano.

G. Casella, R. L. Berger, 2002 (2nd edition), Statistical inference, Duxbury Advanced Series, Wadswarth & Brooks/Cole, Pacific Grove, California.

Teaching methods

Theoretical and practical lessons.

Assessment methods

Written and oral exam.

Teaching tools

Some additional readings concerning the topics which are not included in the recommended textbook will be indicated during the lessons.