29515 - Advanced Statistics

Learning outcomes

By the end of the course, the student is aware of the basic methods for statistical inference both from a Frequentist and Bayesian perspective. More precisely, the student is able to address problems concerning parameter estimation and hypothesis testing within both inferential paradigms.

Course contents

Classical Statistical Inference

• Introduction to classical statistical inference.
  Parametric statistical models. The likelihood function. The exponential family. Statistics and sample moments.
• Estimation theory.
  The sufficiency principle. The likelihood principle. Finite-sample and asymptotic properties of maximum likelihood estimators.
• Hypothesis testing.
  Introduction to hypothesis testing. Frequentist interpretation of the p-value. Uniformly most powerful tests. The likelihood-ratio test. Tests concerning means, variances, proportions, and differences between means.
• Interval estimation.
  Relationships between hypothesis testing and interval estimation. Construction of confidence intervals: frequentist interpretation of the confidence level. Asymptotic confidence intervals for the mean. Confidence intervals for the variance of a normal population.

Bayesian Statistical Inference

• Introduction to Bayesian inference.
  Likelihood, prior distributions, and posterior distributions. Summaries of the posterior distribution. Choice of prior distribution: conjugate and noninformative priors.
• Main univariate models: beta–binomial, gamma–Poisson, and normal–normal models.
• The Bayes factor for hypothesis testing and model selection.
• Markov chain Monte Carlo methods for simulating the posterior distribution and their implementation using Stan in R.
• Inference and predictions based on simulated posterior distributions.
• The Bayesian linear regression model.
• Analysis of case studies in finance and insurance.

 

Readings/Bibliography

Textbooks:

• Azzalini A. (2001) Inferenza Statistica. Una presentazione basata sul concetto di verosimiglianza. Springer.
• Johnson A. A., Ott M. Q., Dogucu M. (2022). Bayes Rules! An Introduction to Applied Bayesian Modeling. Chapman and Hall/CRC, Cap. 1-10. https://www.bayesrulesbook.com
  

Supplementary readings:

• Piccolo D. (2010) Statistica. Il Mulino.
• Held L., Bové S. D. (2014). Applied Statistical Inference: Likelihood and Bayes. Springer.
• Matsuura K. (2023). Bayesian Statistical Modeling with Stan, R, and Python. Springer.
  

Additional teaching materials (slides, exercises, etc.) prepared by the instructors are available on Virtuale.

Teaching methods

• Lectures.
• Practical classes and laboratory sessions in which real-world data will be analyzed using R and Stan software.

Although attendance is not mandatory, active participation in classes is strongly recommended.

Given the type of activities and teaching methods used, attendance at this training activity requires all students to first complete modules 1 and 2 of the e-learning training course on safety in the workplace.

Assessment methods

The examination is designed to assess whether students have achieved the following learning objectives:
- thorough knowledge of the concepts, methods, and tools of advanced statistics covered in the course;
- ability to apply advanced statistical techniques to the critical analysis of datasets;
- ability to interpret the results correctly and communicate them clearly and rigorously.

Module 1

Assessment for Module 1 is conducted as follows.

• Partial exam: a two-hour written examination consisting of open-ended questions and exercises designed to assess students’ understanding of both theoretical and applied course content.
• Full examination: a one-hour written examination consisting of exercises, followed by an oral examination. Admission to the oral examination is conditional on passing the written examination.

During the written examination, students may consult only notes that they have prepared themselves. The notes must be handwritten and may include formulas, descriptions, and any other material considered useful for the examination. They must be contained on a single double-sided A4 sheet. No other materials may be consulted.

The use of AI during assessments is prohibited. Any such use constitutes a violation of academic integrity.

Module 2

The assessment for Module 2 is the same for both the partial and full examinations and consists of two parts:

• Data analysis project: preparation of a report on the application of Bayesian statistical methods and models to a real-world case study using R and Stan. The report must describe the problem under investigation, the data used, the methods adopted, the results obtained, and the corresponding inferential conclusions. The project will be discussed during the examination.
• Oral exam: an oral discussion covering the course content and designed to assess students’ understanding of the main theoretical, methodological, and applied concepts addressed in the course.

The use of AI in the data analysis project is permitted, provided that it is explicitly disclosed, limited in scope, and used only as a support tool. It may not replace the student’s original and critical contribution.

Assessment criteria: grades will be awarded according to the following scale:
<18 insufficient
18–23 sufficient
24–26 fair
27–28 good
29–30 very good
30 with honours excellent

Students with specific learning disorders or temporary or permanent disabilities: students are advised to contact the University's dedicated support office in good time. The office will propose any appropriate accommodations for eligible students. These accommodations must be submitted to the course instructor for approval at least 15 days in advance. The instructor will assess their appropriateness, taking into account the intended learning outcomes of the course.

Examination registration: all students must register for the exam through the AlmaEsami platform in accordance with the general rules established by the University.

Office hours

See the website of Luca Scrucca

See the website of Aldo Gardini