STATISTICS
- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Teaching methods
- Contacts/Info
Essential prerequisite in order to follow the course with profit is the mastery of the topics covered in the following courses
Algebra lineare (I year Mathematics)
Analisi 1, 2 e 3 (I and II year of degree in Mathematics)
Probabilità (II year mathematics)
There is only a final exam, which ensures the acquisition of knowledge through a written test and an oral test.
Written exam will have duration of two hours and a half, without using notes or books, tables (where necessary for the conduct of the written test will be provided along with the exam text); the test consists of two exercises, divided in multiple points, and an application of theory. For each are awarded 12 points and for the question of theory 6 points, to be admitted to the oral test is necessary to achieve the minimum score of 18, of which at least 4 points for the question of theory.
After written exam correction, students who have achieved sufficiency are asked to support the oral examination. This is structured as follows:
- a review of the written exam in which you explain the fixes, you receive any student details and decide whether to change the written exam;
- an oral examination, in order to check the knowledge concerning the notions presented in class, the ability to summarize this knowledge, as well as the ability to solve theoretical and/or practical statistical problems of the types addressed in the course
The assessment of the exam will take particular account of the following parameters:
- ability to organize knowledge discursively;
- rigor and argumentative originality;
- critical reasoning skills on the study carried out and depth of analysis;
- quality of the exhibition, competence in the use of specialist vocabulary, also in relation to its effectiveness and display linearity
It is planned to assign to the oral test at most 10 points in positive or negative.
Educational objectives
The student will have to learn the basic concepts of frequentist statistics and he will be able to handle the tools learned in the course.
The aforementioned objectives aim to prepare the student for the use of statistical probabilistic methods for data analysis and the prediction of the behavior of complex systems as proposed in the objectives of the degree course and for the formation of the professional profile expected for the student in mathematics
Expected learning outcomes
At the end of the course, the student must be able to
1. Introduce probabilistic/statistical models capable of representing real situations;
2. use statistics to condense the information contained in a statistical sample;
3. choose appropriate estimators, punctual or interval, to solve parametric estimation problems or verify the veracity of hypotheses on the distribution that generated the data;
4. apply linear models to the study of dependence between a statistical variable and one or more exogenous variables
Bayesian approach to inferences is becoming increasingly important in various areas of statistics. The aim of the course is to introduce this approach to prametrical statistical inference problem. The main topics are: recalls and additions on probability theory; introduction to statistical survey methods; the Bayesian paradigm and bayesian statistical models; methods for assigning a priori distributions; hierarchical models; Gaussian hierarchical models and variance analysis; some asymptotic results; linear regression model and multiple linear regression.
1. Probability recalls and complements
a. Particular families of univariate probability distributions;
b. References on random vectors: distributions, moments and moment generating functions;
c. Particular families of multivariate probability distributions, in particular Gaussian multivariate distribution;
d. Distributions of transformations of variables and random vectors.
2. Statistical models
a. Samples and sampling methods;
b. The statistics: definition, the sample average, the order statistics. "
c. Examples
3. Point estimation
a. The problem of point estimation and the definition of point estimator;
b. Some methods of researching estimators: the method of moments and the maximum likelihood;
c. Properties of point estimators on finished samples: mean square error, unbiased estimators and "closeness";
d. Asymptotic properties of point estimators: consistency and asymptotic normality;
e. The undistorted estimators in the uniparametric case: the minimization of the variance and its lower limit;
f. Generalizations to the multiparametric case.
g. Examples
4. The estimate by intervals
a. The estimate with intervals and sets of confidence: definitions
b. Methods of findings confidence sets: the pivotal quantity and the statistical method
c. Confidence intervals in Gaussian models
5. Test of hypotheses in the parametric case
a. Definition of the Neyman and Pearson problem: null and alternative hypothesis, simple and compound hypotheses. Examples.
b. Optimality criteria in Neyman and Pearson's test theory: the most powerful uniform tests.
c. Neyman-Pearson’s Lemma
d. Hypothesis testing in Gaussian models.
e. The \ Chi ^ 2 test
6. Linear models
a. Definition of a linear model with one or more regressors;
b. The punctual estimate of the parameters;
c. Interval estimation of parameters;
d. Hypothesis tests
7. Introduction to non parametric problems
a. Point estimate of the distribution function and the Glivenko-Cantelli theorem;
b. The Kolmogorov-Smirnov test for the comparison of distributions;
c. Test for the comparison of two distributions.
The course is structured as follows:
• lectures on blackboard or with tablet and projector for a total of 36 hours;
• Exercises: exercise sheets will be assigned to students, who will be able to solve them at home. Finally exercises will be solved in detail in class by the assistant professor
The course is divided into 36 hours of frontal teaching and 42 hours of practice.
Office hours: by appointment writing to the institutional email address of the teacher or contacting him by phone at the institutional telephone number