30408 - ADVANCED MATHEMATICS AND STATISTICS - MODULE 2 (ADVANCED STATISTICAL METHODS)

Solid knowledge of calculus and of basic programming tools in R facilitates students’ understanding of the topics covered during the course.

Data Science has recently emerged as one of the most exciting interdisciplinary research areas both in academia and among practitioners. The unprecedented availability of data is setting a variety of theoretical and computational challenges for statistics and is, thus, fueling novel groundbreaking developments in the field. Researchers and professional data scientists who want to play a leading role in such a new scenario must definitely have a solid mastery of topics in Probability and Statistics. The main aim of the course is to introduce students to intermediate level tools in Probability Theory and Statistical Inference. The first part is devoted to investigating mathematical aspects of probability, with a special emphasis on multivariate distributions and limiting theorems. In the second part students is guided through the methodological core of point estimation (both from a frequentist and Bayesian perspective), interval estimation, hypothesis testing and regression modeling. These theoretical aspects are complemented by an in-depth presentation of elementary simulation and computational techniques that are routinely used to implement most common statistical procedures.

• Review of discrete and continuous random variables.
• Random vectors.
• Transformations of random variables and of random vectors.
• Simulation of random variables.
• Strong laws of large numbers and the central limit theorem.
• Parametric statistical models.
• Parameter estimation: maximum likelihood and Bayesian methods. 
• Hypothesis testing.
• Regression models.

• Deal with intermediate statistical and probabilistic tools that lie at the foundations of modern Data Science applications.
• Develop a multivariable thinking that is essential to understand and model large and complex datasets.
• Identify drawbacks and merits of both the frequentist and the Bayesian approaches to statistical inference.
• Profitably attend courses on advanced topics in Probability and Stochastic Processes, Statistics and Machine Learning.

• Tailor statistical models to specific experiments, with the aim of addressing estimation and hypothesis testing problems.
• Study relationships among multivariate data, with the aim of drawing predictions and impacting decision-making processes.
• Interpret the output of basic statistical procedures in view of actual applications to real data.

• Face-to-face lectures

• A general written exam or two partial written exams (one in the middle and one at the end of the course).
• There are no different assessment methods or exam programs between attending and non-attending students.

• F.J. SAMANIEGO, Stochastic Modeling and Mathematical Statistics, Boca Raton, FL, CRC Press, 2014.

• M. LAVINE, Introduction to Statistical Thought, 2013, pdf file of the book available at the webpage: http://people.math.umass.edu/~lavine/Book/book.html.