An introduction to the classical theory of statistical models (model identification and estimation, parametric and not parametric models, exponential models), point estimation (moment method, likelihood method and invariant estimators) and methods of evaluating estimators (UMVUE estimators, Fisher information, Cramer-Rao inequality).
This course is designed to introduce the core definitions and concepts of classical mathematical statistics, including statistical models, point estimators, and various estimation methods (such as method of moments, maximum likelihood, and invariance principles), as well as criteria for evaluating estimator performance.
At the end of the course students will be able to
Probability and Statistical Inference
Combination of traditional lectures and exercises.
Review of essential probability including the notion of conditional probability and multivariate normal distribution.
Statistical models and statistics|: the ideas of data sample and of statistical model, identifiability and regular models, the exponential family. Statistics and their distributions. Sufficient, minimal and sufficient, ancillary, complete statistics. The lemma of Neyman-Fisher. The Basu theorem.
Point estimators and their properties: methods to find point estimators: moment methods, least square method, maximum likelihood method, invariant estimators. Methods to evaluate estimators: theorems of Rao-Blackwell and Lehmann-Scheffé. UMVU estimators. Expected Fisher information, Cramer-Rao inequality and efficient estimators.
Statistical hypothesis testing: theorem of Neyman-Pearson for simple hypothesis, likelihood ration test.
Introduction to Bayesian statistics: prior and posterior probability distributions, conjugate priors, improper and flat priors, comparison with the frequentist approach to estimation.
At most one of the last two topics is part of the course for each given year.
Suggested textbooks
G. Casella e R.L. Berger, Statistical inference, Wadsworth 62-2002-02/09
D. A. Freedman, Statistical Models, Theory and Practice, Cambridge 62-2009-05
L. Pace e A. Salvan, Teoria della statistica, CEDAM 62-1996-01 D. Dacunha-Castelle e M. Duflo, Probabilites et Statistiques, Masson 60-1982-18/19/26 e 60-1983-22/23/24 A.C. Davison. Statistical Models, Cambridge University Press, Cambridge, 2003
R. Sundberg, Statistical modelling by exponential families, Cambridge University Press, 2019 B. Efron, Exponential families in theory and practice, Cambridge University Press, 2023 L.D. Brown, Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory, Institute of Mathematical Statistics, 1989
Suggested readings:
D.J. Hand, A very short introduction to Statistics, Oxford 62-2008-05 M. Gasparini, Modelli probabilistici e statistici, CLUT 60-2006-08 S.L. Lauritzen, Graphical models, Oxford University press 62-1996-14 J. Protter, Probability Essentials, Springer 60-2004-09 L. Wasserman. All of Statistics, Springer D. Williams, Probability with Martingales, Cambridge Mathematical Textbooks, 1991
Appunti del docente su aulaweb/Handouts
Ricevimento: For organizational issues contact by email Eva Riccomagno <eva.riccomagno@unige.it>
September 22, 2025
The timetable for this course is available here: EasyAcademy
Written and oral exam.
In the written exam there are three or four exercises. Past exams with solutions are available on the websites.
The written test evaluates the level of learning of the subject, the ability to apply the theory in solving exercises. The oral test evaluates the ability to present, understand and rework the theoretical aspects of the subject.
Students who have valid certification of physical or learning disabilities on file with the University and who wish to discuss possible accommodations or other circumstances regarding lectures, coursework and exams, should speak both with the instructor and with Professor Sergio Di Domizio (sergio.didomizio@unige.it), the Department’s disability liaison.
Upon request by the students, the lectures and/or the exam can be held in English
Prerequisite for the first part: Mathematical Analysis 1 and 2, Probability and Statistical Inference
