OVERVIEW

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).

AIMS AND CONTENT

LEARNING OUTCOMES

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.

AIMS AND LEARNING OUTCOMES

At the end of the course students will be able to

• recognise estimation problems (both parametric and non parametric) in applied contexts
• formulate them in a rigorous mathematical framework
• determine estimators of model parameters and evaluate their goodness
• write definitions, statements and demonstrations and produce related examples and counterexamples

PREREQUISITES

Probability and Statistical Inference

TEACHING METHODS

Combination of traditional lectures and exercises.

SYLLABUS/CONTENT

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.

RECOMMENDED READING/BIBLIOGRAPHY

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 

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

September 22, 2025

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Written and oral exam. 

ASSESSMENT METHODS

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.

FURTHER INFORMATION

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