Vai al contenuto principale

Statistics for stochastic processes


Statistics for stochastic processes


Academic year 2016/2017

Course ID
Teaching staff
Prof. Elvira Di Nardo
Joseph Rinott
1st year
Teaching period
Second semester
D.M. 270 TAF B - Distinctive
Course disciplinary sector (SSD)
MAT/06 - probabilita' e statistica matematica
Formal authority
Type of examination
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 (some introductory material on this topic is present in the text books)

Sommario del corso


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.

Time series are considered, aiming to characterize properties, asymptotic behavior, estimations and forecasting,  spectral analysis 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.


Results of learning outcomes

At the end of the course, students will have understood how to model time series 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



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.


Learning assessment methods

Who wants to be examined on the syllabus of 

a.a.<2015/16: send an e-mail to Elvira Di Nardo, one week before the practical session, to organize the methods

a.a.=2015/16: a practical session on the analysis of a dataset in the computer lab is followed by writing a short essay on one of the arguments introduced  by Prof. Sirovich. The final evaluation with a regular oral examination will be after the correction of this essay and the analysis in the computer lab a couple of days later.

a.a.=2016/17: a practical session on the analysis of a dataset in the computer lab is followed by writing a short essay on one of the arguments introduced  by Prof. Rinott. The final evaluation with a regular oral examination will be after the correction of this essay and the analysis in the computer lab a couple of days later.


Support activities

Computer lab.



1. 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.

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

3. A 16 hours module, included in the courseload, will be taught by Visiting Professor Yosef Rinott on Spectral theory for time series and asymptotics  including  generalization of limit theorems for stationary time series: strongly  and weakly stationary processes, spectral decompositions, ergodic theorems.  

Suggested readings and bibliography


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.

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, 1981

- Shumway and Stoffer, Time series Analysis and Its Applications, Springer, 2011.

For the Lab,  refer to


Class schedule

Last update: 15/05/2017 07:29
Non cliccare qui!