In this course, we will first recall some fundamental concepts of probability calculus. In particular, the idea of a random variable (continuous and discrete) of its density. We will then introduce the vectors of random variables and their joint density to discuss the concept of independence in statistics.
The Law of Large Numbers and the Central Limit Theorem will be explained and commented on.
After recalling the concept of estimation, the maximum likelihood estimation will be defined, and its properties will be discussed. In particular, the intervals and asymptotic tests obtained with the maximum likelihood method will be described.
Prerequisiti
Good knowledge of the fundamentals of Statistics (i.e. descriptive statistics, probability, inferential statistics, simple linear regression model).
Metodi didattici
The course consists of class lectures. The lectures calendar will be published at the beginning of the course on the Moodle e-learning platform.
Verifica Apprendimento
The exam consists of a test including exercises and some open-ended and T/F questions (concerning theoretical topics or short applications of the studied methods)
Contenuti
Part I - Probability (Recap) Discrete and continuous random variables. Density function and probability mass function. Some relevant model (e.g. Bernoulli, Binomial, Poisson, Gaussian, Gamma). Vectors of random variables joint, marginal and conditional density. Independence. Law of Large Numbers and Central Limit Theorem.
Part II - Inference based on the likelihood
Introduction and overview. Statistical inference Sampling Statistics and probability. Some typical problems. Statistics and real problems. Likelihood. Statistical models. Statistical likelihood. Maximum likelihood estimation. Fisher information.
Part III Confidence Intervals and Hypothesis Testing
General aspects. Three test statistics related to the likelihood. Asymptotic properties of the Maximum likelihood estimators. Asymptotic intervals and asymptotic test.
