Statistical inference

PROF. GUIDO CONSONNI

Course aims and intended learning outcomes The course represents a thorough study of principles and methods of statistical inference. It presents in detail the frequentist inference based on the likelihood, but also deals with Bayesian inference which combines the likelihood with the prior. The ultimate aim is to provide students with concepts and methods to navigate through all kinds of statistical analyses they will meet in their professional life. Some aspects will be illustrated using the free and open-source language and environment R. The following learning abilities are provided and expected to be achieved by participants at the end of the course: - Knowledge and understanding Solid knowledge of the foundations and methods of frequentist inference; good grounding in the foundations and methods of Bayesian inference. Knowledge of concepts and properties of: likelihood, statistical information, Bayesian inference, model selection. - Applying knowledge and understanding Ability to understand and interpret the assumptions behind models for empirical analyses, to evaluate the statistical properties of the methods, and to understand their implications from a statistical perspective. - Making judgements Appreciation of uncertainty of judgments based on statistical analyses. Choosing between models in terms of fit and parsimony. - Communication skills Ability to describe with an appropriate statistical language a model used in the analysis and to communicate the results of empirical findings using appropriate summaries. - Lifelong learning skills Skills for data-analysis and statistical learning need understanding of the methods in order to learn proficiently future developments, in Statistics and Data Science.

Course content – Likelihood and log-likelihood; maximum likelihood (ML) estimation; score function; sufficiency. – Frequentist inference: unbiasedness, consistency, standard errors and confidence intervals; significance tests and p-values. – Frequentist properties of the likelihood; expected Fisher information; score statistic; distribution of the ML estimator and the Wald statistic. – Likelihood ratio statistic. – Likelihood inference in multiparameter models. – Bayesian inference: Bayes’ theorem, prior and posterior distribution, improper priors, loss function and Bayes estimate. – Model selection: Akaike’s Information Criterion (AIC); Cross validation and AIC; Bayesian Information Criterion (BIC).

Reading list L. HELD-D. SABANÉS-BOVÉ, Aookied Statistical Inference. Likelihood and Bayes, Second edition, Springer, 2019. Language and environment: R - http://www.r-project.org/ Class notes, coding and further material will be posted on the University platform Blackboard.

Teaching method Lectures (60 hours) complemented with exercise sessions (20 hours) and an introduction to R.

Assessment method and criteria Written examination with questions on methods and exercises. Students enrolling in this course are expected to know data analysis, probability and frequentist inference, at the level of Statistics courses usually taught in a bachelor degree in Economics; see for instance the topics covered in ‘Statistica (analisi dei dati e probabilità) and ‘Statistica applicata’ (or ‘Statistics’ and ‘Applied Statistics’) at this University. These topics will be presented in a preliminary course in Statistics which will be held in the week before the start of the lectures. Students whose knowledge of statistical inference is weak are suggested to attend this course.

Notes and prerequisites Prerequisites Students enrolling in this course are expected to know data analysis, probability and frequentist inference at the level of Statistics courses usually taught in a bachelor degree in Economics. (More specific Notes at the bottom).