LEARNING OUTCOMES

To formalise estimation problems (parametric and non-parametric) and statistical hypothesis testing in a rigorous mathematical framework. To intruduce the analysis of time series, combining practical and theorical considerations.

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

At the end of the course students will

• be able to perform the analysis of simple time series in the time domain also with software
• be able to develop further theoretical and computational knowledge for statistical analysis of time series
• be able to present a simple report about the statistical analysis of a time series
• possess the essential mathematical and statistical knowledge related to time series

TEACHING METHODS

Combination of traditionals lectures and, only for the second part, lab sessions with the software R.

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.

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.

Program of the second part of the course:
Time series: exploratory analysis. The notions of stationarity and ergodicity. Strong and weak stationary processes. Autocovariance function and partial autocovariance function. SARIMA models.

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 

Second Part: 
C. Chatfield (1980). The analysis of Time Series: an introduction, Chapman and Hall
Rob J Hyndman and George Athanasopoulos, Forecasting: Principles and Practice, Monash University, Australia https://otexts.com/fpp2/
R.D. Pend , F. Dominici, Statistical methods for environmental epidemiology with R. A case study in air pollution and Health
R.H. Shumway, D.S. Stoffe, Time series analysis and its applications with examples in R