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
• identify suitable regression models for data analysis, to analyse the data with advanced software, to summarise results of the analysis in a report including the interpretation of the results of the analysis and of their reliability.

TEACHING METHODS

Classroom lectures and exercise sessions. In the second part there will be computer laboratory sessions whose aim is to practice the application of the theoretical models learnt during classroom lectures, to describe and predict a phenomenon of interests based on real case studies and data sets. During the lab sessions the student will be able to verify his/her level on understanding of the theory and its application.

SYLLABUS/CONTENT

Program of the first part of the course:
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.

Program of the second part of the course:
General linear models. ANOVA: crossed and nested factors; unbalanced data. Overparametrised models: reparametrization and generalised inverse function: theoretical considerations and practical implications. Multivariate linear regression models and models for repeated measures. 

Generalised linear model. Exponential family. Link function. Models for categorical data (binomial, multinomial and Poisson models). Iterative methods for coefficients’ estimation: Newton-Raphson, scoring. Asymptotic distributions for likelihood based statistics. Statistical hypothesis testing and goodness of fit criteria: deviance, chi-squared. Residuals. Tests and confidence intervals for (subsets of) the models parameters. Odds-ratio and log-odd ratios. Models for ordinal data and contingency tables. 

Lab sessions based on the softwares SAS and R. 

RECOMMENDED READING/BIBLIOGRAPHY

Prima Parte/First part:

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 

Seconda Parte/Second part:

Dobson A. J. (2001). An Introduction to Generalized Linear Models 2nd Edition. Chapman and Hall.
Rogantin M.P. (2010). Modelli lineari generali e generalizzati. In rete.