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Course Topics
M1 - Review of Probability: Probability spaces, random variables, distributions; expectation, variance, covariance; common discrete and continuous families. - Multivariate Distributions and Transformations: Joint, marginal, and conditional distributions; independence; functions of random variables. - Sampling Distributions: Random samples; sample mean and sample variance; normal-theory results; chi-square, t, and F distributions. - Principles of Estimation: Point estimators; bias, variance, and mean squared error; method of moments; introduction to sufficiency. - Maximum Likelihood Estimation: Likelihood functions; MLEs for common models; invariance property; numerical optimization ideas. - Properties of Estimators: Unbiasedness and consistency; efficiency; Cramér–Rao lower bound; Rao–Blackwell theorem (introductory treatment). - Confidence Intervals: Pivotal quantities; exact and approximate confidence intervals; confidence intervals based on asymptotic normality. - Hypothesis Testing: Null and alternative hypotheses; Type I and Type II errors, power; Neyman–Pearson lemma; likelihood ratio tests. - Bayesian and Likelihood Perspectives: Likelihood principle; basic Bayesian inference; posterior distributions and credible intervals; comparison with classical inference. - Asymptotic Inference: Law of large numbers; central limit theorem; delta method; asymptotic distribution of MLE. - Modern and Computational Topics: Bootstrap methods; permutation tests; Monte Carlo methods; extension of the likelihood function (penalized, M-estimation) M2 - Flexible Linear Regression: Recap of linear regression and diagnostics; polynomial regression, interactions, and categorical predictors; splines and local regression; flexibility-interpretability trade-off. - Classification: Logistic regression; linear discriminant analysis and comparison with logistic regression; quadratic discriminant analysis and k-nearest neighbours; classifier evaluation through confusion matrix, ROC curves, and AUC. - Model Selection and Assessment: Training and test error; bias-variance trade-off; validation set approach, k-fold and leave-one-out cross-validation; best subset and stepwise selection; information criteria. - Clustering: k-means and hierarchical clustering; linkage criteria and dendrograms; introduction to model-based clustering and mixture models; notes on density-based approaches. - Dimensionality Reduction: Principal component analysis; exploratory factor analysis as an alternative perspective; considerations on high-dimensional settings. - Generalized Linear Models: Exponential family and link functions; GLM formulation; logistic regression revisited as a GLM; Poisson regression for count data; estimation, deviance, and goodness of fit.

Teaching format
Recorded lectures, in-person teaching, exercises. The course adopts a blended, student-centred approach that emphasises problem-based learning and active engagement. A portion of the lecture content is made available online in advance, allowing students to explore key concepts independently and at their own pace before attending class. This preparatory work enables inperson sessions to focus on the application of knowledge through real-world problems, collaborative activities, and guided discussions — fostering critical thinking and deeper learning. The course is fully aligned with the principles of the Italian Universities Digital Hub (EDUNEXT) initiative (https://edunext.eu), which promotes the integration of digital resources and active learning strategies within university teaching.

Educational objectives
Intended Learning Outcomes (ILO) M1: ILO 1 Knowledge and understanding: ILO 1.1 The student acquires knowledge of the analytical techniques and tools required to understand and quantitatively analyse economic and business phenomena in order to support decision-making processes. ILO 1.2 The student consolidates knowledge of statistical inference, linear models and their generalisations, linear algebra, and optimisation techniques. ILO 1.3 The student acquires an in-depth knowledge of the main techniques of supervised and unsupervised statistical learning, which are instrumental in the development of analysis and visualisation of economic and business data. ILO 2 Applying knowledge and understanding: ILO 2.1 Ability to apply and implement analysis techniques focusing on different types of datasets such as streaming data, tabular data, documents and images and analysis on joint datasets. ILO 2.2 Ability to apply supervised and unsupervised learning, and knowledge modelling, extraction, integration, analysis and exploitation; these skills are declined in various application domains of interest to companies and public and private organisations. ILO 3 Making judgements: ILO 3.1 The student acquires the ability to apply acquired knowledge to interpret data in order to make directional and operational decisions in a business context. ILO 3.2 The student acquires the ability to apply acquired knowledge to support processes related to production, management and risk promotion activities and investment choices through the organisation, analysis and interpretation of complex databases. ILO4 Communication skills: ILO 4.1 The student acquires the ability to communicate effectively in oral and written form the specialised content of the individual disciplines, using different registers, depending on the recipients and the communicative and didactic purposes, and to evaluate the formative effects of his/her communication. ILO 5 Learning skills: ILO 5.1 The student acquires knowledge of scientific research tools. He/she will also be able to make autonomous use of information technology to carry out bibliographic research and investigations both for his/her own training and for further education. Furthermore, through the curricular teaching and the activities related to the preparation of the final thesis, she will be able to acquire the ability - to identify thematic connections and to establish relationships between methods of analysis and application contexts; - to frame a new problem in a systematic manner and to implement appropriate analysis solutions; - to formulate general statistical-econometric models from the phenomena studied. M2: ILO 1 Knowledge and understanding: ILO 1.1 The student acquires knowledge of the analytical techniques and tools required to understand and quantitatively analyse economic and business phenomena in order to support decision-making processes. ILO 1.2 The student consolidates knowledge of statistical inference, linear models and their generalisations, linear algebra, and optimisation techniques. ILO 1.3 The student acquires an in-depth knowledge of the main techniques of supervised and unsupervised statistical learning, which are instrumental in the development of analysis and visualisation of economic and business data. ILO 2 Applying knowledge and understanding: ILO 2.1 Ability to apply and implement analysis techniques focusing on different types of datasets such as streaming data, tabular data, documents and images and analysis on joint datasets. ILO 2.2 Ability to apply supervised and unsupervised learning, and knowledge modelling, extraction, integration, analysis and exploitation; these skills are declined in various application domains of interest to companies and public and private organisations. ILO 3 Making judgements: ILO 3.1 The student acquires the ability to apply acquired knowledge to interpret data in order to make directional and operational decisions in a business context. ILO 3.2 The student acquires the ability to apply acquired knowledge to support processes related to production, management and risk promotion activities and investment choices through the organisation, analysis and interpretation of complex databases. ILO4 Communication skills: ILO 4.1 The student acquires the ability to communicate effectively in oral and written form the specialised content of the individual disciplines, using different registers, depending on the recipients and the communicative and didactic purposes, and to evaluate the formative effects of his/her communication. ILO 5 Learning skills: ILO 5.1 The student acquires knowledge of scientific research tools. He/she will also be able to make autonomous use of information technology to carry out bibliographic research and investigations both for his/her own training and for further education. Furthermore, through the curricular teaching and the activities related to the preparation of the final thesis, she will be able to acquire the ability - to identify thematic connections and to establish relationships between methods of analysis and application contexts; - to frame a new problem in a systematic manner and to implement appropriate analysis solutions; - to formulate general statistical-econometric models from the phenomena studied.

Assessment
The overall exam mark will be determined by the assessment of the two modules (M1+M2). M1 Final exam (60%)(ILO 1.1–1.3, 2.1, 2.2, 4.1) and 4 homework assignments (10% each, 40% total)(ILO 2.1, 2.2, 3.1, 3.2, 5.1). Students who do not submit the homework will be assessed entirely on the final exam (100%). M2 Final exam (60%)(ILO 1.1-1.3, 2.1, 2.2, 5.1) and homeworks/data analysis assignments to be handed in (40%)(ILO 2.1, 2.2, 3.1, 3.2, 4.1). Students who do not complete the homeworks/assignment will be assessed entirely on the final exam (100%)

Evaluation criteria
In both modules the exam modalities are the same for the attending and the non-attending students. Project work (40% of the final grade) and written exam (60% of the final grade). - Written exam: M1: understanding of probabilistic and inferential concepts, correct derivation and interpretation of estimators and tests, clarity and precision of explanations M2: understanding of statistical concepts, correct interpretation of the results of statistical analyses, clarity and precision of explanations - Project work: clarity of presentation, ability to gain useful and novel insights from data, creativity and critical thinking, adherence to reproducible research practices, proficiency in using R and other software to perform data preparation, statistical analyses, and graphical representations appropriate to the data at hand.

Required readings

M1

• Rice, J. A. (2007) Mathematical Statistics and Data Analysis. Cengage Learning.
• Hogg, R. V., McKean, J. W., and Craig, A. T. (2019) Introduction to Mathematical Statistics. Pearson.
• Casella, G., & Berger, R. (2024). Statistical inference. Chapman and Hall/CRC.

M2

• Hastie, T., Tibshirani, R., & Friedman, J. (2009). The elements of statistical learning. Springer
• James, G., Witten, D., Hastie, T., Tibshirani, R. (2013) An Introduction to Statistical Learning with Applications in R. Springer.

Supplementary readings

M2

• Bishop, C. M. (2006). Pattern recognition and machine learning. New York: Springer.
• Slides and lecture notes provided

Further information
M1 & M2 - Software Students may use R, Python or another computer program of their choice for computational assignments. Example code and demonstrations may be provided in R.