# MATHEMATICAL STATISTICS

7 credits during the 1st year of 9011 Mathematics (LM-40) GENOVA

7 credits during the 2nd year of 9011 Mathematics (LM-40) GENOVA

**Mathematical Statistics and Data Management 8766 (coorte 2020/2021)**- PROBABILITY 87081

**Mathematical Statistics and Data Management 8766 (coorte 2019/2020)**- PROBABILITY 87081

**Mathematical Statistics and Data Management 8766 (coorte 2018/2019)**- PROBABILITY 87081

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).

The second part develops the main aspects of the theory and practice of the analysis of time series in the time-domain and hints to the analysis in the frequency domain.

## AIMS AND CONTENT

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** **

## TEACHERS AND EXAM BOARD

**Ricevimento:** By appointment arranged by email with Luca Oneto luca.oneto@unige.it and Fabrizio Malfanti <fabrizio.malfanti@intelligrate.it>
For organizational issues contact by email Eva Riccomagno <riccomagno@dima.unige.it>

Exam Board

EVA RICCOMAGNO (President)

MARTA NAI RUSCONE

MARIA PIERA ROGANTIN (President Substitute)

## LESSONS

Teaching methods

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

LESSONS START

The class will start according to the academic calendar.

## EXAMS

Exam description

The two parts of the course are examined together. Written and oral exam. For the second part written exam with multiple choice and open questions. Two group projects on topics agreed with the teachers. Discussion of the reports and written test.

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).

Main points of evaluation are the level of acquisition of the learning objectives and the ability to communicate in a written report the data analyzes carried out during the course.

Exam schedule

Date | Time | Location | Type | Notes |
---|---|---|---|---|

04/06/2021 | 09:00 | GENOVA | Scritto | |

04/06/2021 | 09:00 | GENOVA | Orale | |

09/07/2021 | 09:00 | GENOVA | Scritto | |

09/07/2021 | 09:00 | GENOVA | Orale | |

06/09/2021 | 09:00 | GENOVA | Scritto | |

06/09/2021 | 09:00 | GENOVA | Orale |

## FURTHER INFORMATION

**Pagina web dell'insegnamento:**

Prima parte: http://www.dima.unige.it/~riccomag/Teaching/StatisticaMatematica.html

Seconda parte: http://www.dima.unige.it/~rogantin/ModStat/

**Prerequisiti Prima Parte:** Analisi Matematica I e 2. Calcolo delle Probabilità .

**Prerequisiti Seconda Parte:** Argomenti di Statistica inferenziale e della prima parte di Statistica Matematica (quest'ultima svolta in parallelo) con corrispondenti prerequisiti.

**Web pages of the couse are**

for the first part: http://www.dima.unige.it/~riccomag/Teaching/StatisticaMatematica.html

for the second part: http://www.dima.unige.it/~rogantin/ModStat/

**Prerequisite for the first part**: Mathematical Analysis 1 and 2, Probability

**Prerequisite for the second part: **Statistical inference and in parallel the first part of Mathematical Statistics.