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

To formalise estimation problems (parametric and non-parametric) and statistical hypothesis testing in a rigorous mathematical framework, to formulate and apply appropriate regression models to various typologies of data sets.

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, Mathematical Analysis 1 and 2

TEACHING METHODS

Combination of traditionals 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.

Some academic years:

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

Testi consigliati/Text books:   

G. Casella e R.L. Berger, Statistical inference, Wadsworth 62-2002-02  62-2002-09
D. A. Freedman, Statistical Models, Theory and Practice, Cambridge 62-2009-05

L. Pace e A. Salvan, Teoria della statistica, CEDAM 62-1996-01   
M. Gasparini, Modelli probabilistici e statistici, CLUT 60-2006-08   
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 

Letture consigliate/Suggested reading:

David J. Hand, A very short introduction to Statistics, Oxford 62-2008-05
L. Wasserman. All of Statistics, Springer 
J. Protter, Probability Essentials, Springer 60-2004-09 
S.L. Lauritzen, Graphical models, Oxford University press 62-1996-14 
D. Williams, Probability with Martingales, Cambridge Mathematical Textbooks, 1991

Appunti distribuiti a lezione/Handouts 

TEACHERS AND EXAM BOARD

Exam Board

EVA RICCOMAGNO (President)

SARA SOMMARIVA

FRANCESCO PORRO (President Substitute)

LESSONS

LESSONS START

The class will start according to the academic calendar. 

Class schedule

MATHEMATICAL STATISTICS

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 oral exam consists of questions on both parts of the course. The course work done during the lab sessions might be subject of the oral exam (thus bring with you at the exams that course work). 

Exam schedule

FURTHER INFORMATION

Students with DSA, disability or other special educational needs are recommended to contact the teacher at the beginning of the course, in order to organize teaching and assessment, taking in account both the class aims and the student's needs and providing suitable compensatory instruments.

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