The main aim of the course is to introduce the basic concepts of univariate Mathematical Statistics, both from a theoretical and applied point of view. Some very first element also of multivariate statistics will be introduced. During the lab activities, the students will be trained to perform simulations and data analyses with advanced software instruments (Matlab, SAS) and to produce suitable reports, using an appropriate technical language.
Expected learning outcomes
Basic notions and theorems of univariate Mathematical Statistics. The student will then be able to apply and broaden his/her knowledge of the subjects in different areas of interest, both in theoretical and applied contexts, and to perform statistical data analyses.
1. Convergence of random variables and properties of estimators 1.1. Sufficiency 1.2. Completeness 1.3. Methods for variance reduction: The Rao-Blackwell and Lehmann-Scheffe' Theorems 1.4. The Cramer-Rao Theorem - Efficiency
2. Testing of Statistical Hypotheses 2.1. Power of a test and UMP tests 2.2. The Neyman-Pearson Lemma 2.3. Likelihood ratio 2.4. Classical parametric tests
3. The general linear model 3.1. Regression analysis 3.2. Analysis of variance
4. Introduction to non parametric methods 4.1. Run test and other nonparametric tests. 4.2. Chi Square tests 4.3. Glivenko-Cantelli Theorem and Kolmogorov-Smirnov test (hints) 4.4. Permutation tests
5. Laboratory of simulation and data analysis (with the use of MATLAB, SAS, R) 5.1. Confidence intervals 5.2. Simulation of random processes 5.2.1. The non homogeneous Poisson process 5.2.2. The spatial Poisson process 5.3. Complements and examples of estimation theory 5.3.1. Density estimation via histograms 5.3.2. Kernel density estimation 5.4. Introduction to Monte Carlo methods 5.5. Introduction to statistical softwares (SAS, R) 5.6. Statistics with software 5.6.1. Descriptive Statistics 5.6.2. Hypothesis testing 5.6.3. General linear model 5.6.4. Analysis of variance
The students of the old bachelor programme who follow the course only for 6 ects may skip the Laboratory program from item 5.1 to 5.4 included.