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 fundamentals of simulations and data analyses will be provieded with advanced software instruments (Matlab, SAS, R or similar software).
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
Prerequisites for admission
Basic course in Probability
Frontal lessons for theory and exercises; computer room for the lab
G. Casella and R.L. Berger, Statistical Inference. Second edition (2001) J. Shao, Mathematical statistics. Second edition (2003)
Bibliography for the lab: S.Iacus, G.Masarotto, Laboratorio di statistica con R, McGraw Hill, 2003 F. Ieva, C. Masci, A.M. Paganoni, Laboratorio di statistica con R (seconda edizione), Pearson, 2016
Assessment methods and Criteria
The final examination consists of three parts: a written exam, an oral exam and a lab exam.
- During the written exam, the student must solve some exercises in the format of open-ended questions, with the aim of assessing the student's ability to solve problems in statistical analysis. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). In place of a single written exam given during the first or the second examination session, the student may choose instead to take 2 midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal. - The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them. -The lab exam consists in developing a project on statistical analysis of data chosen by the student. The project will be presented by the student during the written exam. The lab portion of the final examination serves to assess the capability of the student to put a problem of data analysis with advanced software instruments, and to produce suitable reports, using an appropriate technical language.
The complete final examination is passed if all three parts (written, oral, lab) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.