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. Random sample and statistical models. The exponential family. 2. Properties of estimators: consistency, unbiasedness, asymptotic normality. 3. Methods of finding estimators: 3.1. Maximum likelihood function and maximum likelihood estimators. 3.2. Method of moments. 4. Interval estimation. 5. Hypothesis testing 5.1. Power function and UMP tests 5.2 The Neyman-Pearson Lemma 5.3. Likelihood ratio 5.4 Classical parametric tests 6. Further properties of estimators: 6.1. Sufficiency. 6.2. Completeness. 6.3. Methods for variance reduction: The Rao-Blackwell and Lehmann-Scheffe' Theorems. 6.4. The Cramer-Rao Theorem 6.5. Efficiency and Fisher's information. 7. The general linear model. 8. Introduction to non-parametric models. 9 Laboratory of simulation and data analysis 9.1 Descriptive Statistics 9.2 Complements and examples of estimation theory: density estimations via histograms and kernels. 9.3 Simulation of random processes: the homogenous and non-homogeneous Poisson process. 9.4 Statistics with software: 9.4.1 Confidence intervals 9.4.2 Hypothesis testing 9.4.3 Analysis of variance 5.2.1 Non-parametric tests 9.4.5 Linear regression