Statistics and Data Analytics

Activity type

Statistico, statistico applicato, demografico

SSD

SECS-S/01

Year

N/D

Semester

N/D

CFU

3

Hours distribution

27 classroom hours

Lecturers

CRISTINA MOLLICA

Activity type

Statistico, statistico applicato, demografico

SSD

SECS-S/01

Year

N/D

Semester

N/D

CFU

9

Hours distribution

72 classroom hours

Lecturers

LUCA TARDELLA

Learning goals.
The primary educational objective of the course is students' learning of the main problems and methods of statistical Inference and its different alternative theoretical approaches.
Students must also be able to solve the analytical problems necessary to apply the above methods and be able to interpret the results that derive from their application to real data.

Knowledge and understanding.
After attending the course the students know and understand the main inferential problems (point and interval parametric estimation and hypothesis testing of the most important univariate statistical models) and the main methods to be used to solve these problems (for example: maximum likelihood estimation, confidence intervals, parametric tests).

Applying knowledge and understanding.
At the end of the course the students are able to formalize real problems in terms of inferential problems and to apply the specific methods of the discipline to solve them.
They are also able to process the most important statistical models (with one or two unknown parameters) and to apply the methods to models not covered in the lessons.
Finally, they are able to apply the methods to the data and to interpret the results.

Making judgements.
Students develop critical skills through the application of inferential methodologies to a wide range of statistical models.
They also develop the critical sense through the comparison between alternative solutions to the same problem obtained using different inferential logics.
They learn to critically interpret the results obtained by applying the procedures to real data sets.

Communication skills.
Students acquire by means of theoretical study and by solving practical exercises, the technical-scientific language of the discipline, which must be properly used both in the intermediate and final written tests and in the oral axam.
Communication skills are also developed through group activities stimulated during labs and participation to a public discussion forum Learning skills.

Students who pass the exam have learned a method of analysis that allows them to tackle, in future more advanced courses, the study of the formal properties of inferential procedures in more complex modeling contexts.

Module: LABORATORY
N/D
Module: STATISTICAL INFERENCE
N/D

Students must possess basic knowledge of mathematical analysis (in particular, all the analytical tools for studying a real function of a real variable; derivatives and integrals for real functions of real variables) and probability (in particular, random variables, probability distributions, moments, and convergence of sequences of random variables). This knowledge is acquired by passing the exams in the Mathematics (Course II) and Probability courses.

Module: LABORATORY
N/D
Module: STATISTICAL INFERENCE
This course can be divided in 5 main parts:

- Introductory examples of statistical inference problems in Science and Society; formalization of inferential problems [1.] (10 hours)
- Likelihood function and inference based on the likelihood principle [2.] (10 hours)
- Repeated sampling principle, sampling distributions and frequentist inference for point estimations [3.] and interval estimation [4.] (26 hours)
- Frequentist inference for testing hypothesis [5.] (20 hours)
- Brief introduction to Bayesian inference [6.] (6 hours)

The students are exposed to further examples and insights by means of 12 weekly 2-hour computer lab sessions with the R software [open source]. (24 ore)

(further information can be found on the course website hosted on the e-learning Moodle Platform - http://elearning.sta.uniroma1.it/moodle2) [https://elearning2.uniroma1.it/course/view.php?id=14043]

Detailed list of topics:

1. Probability laws and parametric families of distributions. Random variable and stochastic model for observable phenomena. Simple Random sampling and statistical model for n observations. Statistical inference problems setup. Overview of the main inferential problems.

2. Likelihood function and inference based on likelihood function. Maximum likelihood estimate. Set-valued estimation and level sets. Likelihood ratio. Observed Fisher information. Normal approximation of the likelihood function. Approximate level sets. Likelihood principles. Sufficiency. Exponential families.

3. Classical parametric estimation. Distribution of sample statistics. Sample mean and sample variance. General properties of sample mean and sample variance. Distribution of sample mean and sample variance under Gaussian assumptions. Asymptotic theorems: law of large numbers and central limit theorem. Point estimators. Maximum likelihood estimator (MLE). Estimators derived from the method of moments. Criteria for evaluating estimators: MSE (Mean Square Error). Consistency. Unbiased estimators. Optimal estimators and conditions for achieving optimality. Information in a random sample and Cramer-Rao lower bound. Sufficiency and its role in statistical inference. Rao-Blackwell procedure. Asymptotic properties of MLE. Delta method.

4. Set-valued and interval estimation. Confidence intervals: definition and examples. Pivotal quantities. Aproximate confidence intervals based on asymptotic normality of MLE and Delta method.

5. Statistical hypothesis test. General concepts: hypothesis system; alternative formalization of test; different types of error (type I - type II). Error control in tests: simple hypotheses and composite hypotheses. Neyman-Pearson lemma characterizing optimal testS. General system of composite hypotheses: power function. Test optimality. Generalized likelihood ratio test. Asymptotic tests. Hits on non-parametric tests: normality tests and chi-square test.

6. Basic introduction to Bayesian inference: Bayes formula (discrete and continuous). Bayesian statistical model. Prior and posterior distributions. Predictive distribution. Posterior distribution summaries. Point estimates. Set-valued inference: credible intervals (equal-tails) and HPD regions. Hypothesis testing: posterior probability of statistical hypothesis and Bayes Factor (hints). Conjugate analysis for a Bernoulli model. Conjugate analysis for a Gaussian model. Hints to non informative analysis and comparison of alternative statistical frameworks (frequentist vs Bayesian).

COMPUTER LAB

There are 12 lab sessions held in a computer lab with two intermediate tests. These lab sessions are focussed on the following list of topics:

A) Introduction to the R environment. Memory object of the current work session: `numeric` and `function` mode objects. S3 classes. Function object definition.

B) Basics of graphics in R. Interactive graphs and main parametric families of distributions. Probability distributions and statistical models. Simulation from a known statistical model.

C) Likelihood function definition and graphical exploration. R function for Maximum Likelihood estimation.

D) Random variable simulation. Probability calculus, asymptotic results and simulation-based approximation.

E) Repeated sampling (frequentist) principle.Sampling distributions of statistics and estimators. Estimator simulation. Estimator error evaluation with increasing sample size.Comparison between exact and asymptotic distributions of an estimator. Exact distribution and its simulation-based approximation.

F) Mean square error and its approximation. Comparing estimators.

G) Introduction to confidence intervals. R code from scratch and built-in functions and packages for confidence intervals. Asymptotic confidence interval based on MLE.Optimal width for confidence interval. Comparison between two alternative confidence intervals.

H) Hypothesis test for simple hypotheses. Parametric and nonparametric hypothesis test with R. Computing a displaying a power function. P-value. Nonparametric tests on real data

I) Basics for data import and data manipulation.Inferential analysis of real data.

Module: LABORATORY
N/D
Module: STATISTICAL INFERENCE
Suggested textbook:

• "Inferenza Statistica" (edited by F. De Santis, S. Gubbiotti, L. Tardella, I. Verdinelli) lecture notes downloadable from the course website hosted on the elearning Moodle platform (http://elearning.sta.uniroma1.it/moodle2)

• Exercise collection downloadable fro the elearning Moodle platform(http://elearning.sta.uniroma1.it/moodle2)

Module: LABORATORY
N/D
Module: STATISTICAL INFERENCE
- D. Piccolo (2010). Statistica. Il Mulino

Module: LABORATORY
N/D
Module: STATISTICAL INFERENCE
Teaching activities take place in the lecture room and comprise presentation of theoretical aspects, applications to most common univariate statistical models, solution of exercises. Theoretical in person teaching is then complemented with hands-on and interactive learning during computer lab sessions.

Module: LABORATORY
N/D
Module: STATISTICAL INFERENCE
Attendance of the course is optional, but warmly recommended. In case of impossibility to attend lectures and/or labs, it is advisable to contact the teacher immediately. All contact information is provided on the teacher's personal bulletin board which can be retrieved from the study course catalog https://corsidilaurea.uniroma1.it/cerca/docente

Module: LABORATORY
N/D
Module: STATISTICAL INFERENCE
To be able to register the 12 credit exam the student must have passed

- written test
- oral exam
- practical test in the computer laboratory using the R software

The order in which the three tests must be passed is the one indicated. It can only be modified with regard to the practical laboratory test.

All students (attending and not) are allowed to pass the written test by passing a total of two midterm tests.
All students (attending and not) are allowed to pass the practical test of the Laboratory by passing two midterm tests.

The dates of the written tests are scheduled for the entire academic year and correspond to the exam dates published on INFOSTUD.
The dates of the Laboratory tests are scheduled for the entire academic year and correspond to the exam dates published on INFOSTUD.
The midterm tests are usually scheduled with the following criteria: first midterm test in the week starting with Easter Monday.
The second midterm test is scheduled either on the last day of the lesson period or in the immediately following days.

The validity of the tests passed is one calendar year from the date of the exam taken.

Module: LABORATORY
N/D
Module: STATISTICAL INFERENCE
N/D

Module: LABORATORY
N/D
Module: STATISTICAL INFERENCE
N/D