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Department of Mathematics "Giuseppe Peano"

# Laurea Magistrale (M.Sc.) in Stochastics and Data Science

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## STATISTICS FOR STOCHASTICS PROCESSES

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Course ID
MAT0038
Teaching staff
Prof. Roberta Sirovich
Prof. Elvira Di Nardo
Year
1st year
Teaching period
Second semester
Type
D.M. 270 TAF B - Distinctive
Credits/Recognition
6
Course disciplinary sector (SSD)
MAT/06 - probabilita' e statistica matematica
Delivery
Class Lecture
Language
English
Attendance
Optional
Type of examination
Mixed
Prerequisites
Good knowledge of probability theory and the basics of stochastic processes. Some analysis is also required. In more details you will need
- laws of large numbers and central limit theorems
- measure theory
- conditional expectations
- L^p spaces with respect to a probability measure
- Hilbert spaces and projections (some introductory material on this topic is present in the text books)
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## Course objectives

The goal of lectures is to introduce statistical inference for stochastic processes (e.g. time series) taking into account both the theoretical/mathematical aspects and their practical application to data analysis.

We introduce the mathematical tools needed to prove asymptotic properties of estimators such as consistency and asimptotic normality in the framework of stochastic processes.

Then time series are considered, aiming to characterize properties, asymptotic behavior, estimations and forecasting, as well as decomposition in trend and seasonal components. Such concepts are applied to the analysis of simulated data or existing databases in order to infer and validate a model supporting the data.

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## Results of learning outcomes

At the end of the course, students will have understood the role of limit theorems in asymptotic statistics, with focus on forecasting and estimation of the moments, of the spectrum and of the parameters of time series models.

Moreover they will know which are the main steps of the analysis of a dataset, and which tools are available to this aim:
- descriptive statistics, moment and spectrum estimation
- formulation of models, parameter estimation, model selection, model verification
- forecasting

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## Course delivery

We will mainly deliver frontal lectures, but a computer lab is also included. During the lectures we will alternate a formal presentation of some topics, including proofs and technical details, with a more informal part where we will introduce some concepts that are useful for the analysis of data sets. In the computer lab we will use R to simulate and analyse datasets from ARMA processes or existing databases. We refer to some particular packages useful to deal with simulations, decompositions and forecasting.

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## Learning assessment methods

A practical session on the analysis of a dataset in the computer lab is followed by a regular oral examination.

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Computer lab.

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## Program

1. Asymptotic statistics for iid samples: limit theorems applied to large sample statistics. Consistency and asymptotic normality of different estimation procedures (moments, parameters.).

2. Generalization of limit theorems for stationary time series: strongly  and weakly stationary processes, spectral decompositions, ergodic theorems. Central limit theorems for m-dependent processes, linear processes, strongly mixing processes and martingale differences. Statistical applications in moments, spectrum and parameter estimation.

3. Time series: ARMA processes, covariance and spectrum. Estimation and elimination of trend, seasonal components and periodicities. Linear filtering, causality and smoothing. Recursive methods for computing the best linear predictors: Durbin-Levinson algorithm, innovations algorithms. Estimation of the mean, the covariance, the partial autocorrelation. Estimation of the parameters and model selection. Diagnostic tools. Asymptotic theory.  Spectral representation of time series and estimation of the spectrum.

4. Computer lab: simulation and statistical analysis of time series with R.

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Lectures will not adhere to the material of any single text, but the students can find material on the topics we teach on different books. A detailed list of references specialized for each topic will be made available during the course.

Asymptotic statistics for iid samples:
- A.W. van der Vaart, Asymptotic Statistics, Cambridge University Press 1998.
- Ferguson, A course in large sample theory, Chapman & Hall, 1996

Statistical inference for time series
- Brockwell and Davis, Time Series, theory and methods, Springer (collana SSS), New York, 1991
- Priestley, Spectral Analysis and Time Series, Academic Press, Vol I, 1982
- Shiryaev, Probability, Springer (collana GTM 95), New York, 1996
- Shumway and Stoffer, Time series Analysis and Its Applications, Springer, 2011.

For the Lab,  refer to www.stat.pitt.edu/stoffer/tsa3/

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## Class schedule

DaysTimeClassroom
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Last update: 09/05/2016 16:15
Location: https://www.master-sds.unito.it/robots.html