Laurea Magistrale (M.Sc.) in Stochastics and Data Science

Statistical inference

Statistical inference

Academic year 2018/2019

Sommario del corso

• Course objectives
• Results of learning outcomes
• Course delivery
• Learning assessment methods
• Program
• Suggested textbooks and readings
• Teaching tools

Course objectives

Ability to apply statistical concepts and statistical techniques with respect to the point estimation, hyphotesis testing and confidence sets.

Results of learning outcomes

Knowledge and understanding
Advances knowledge of statistical modeling via point estimation, hypothesis testing and confidence intervals.

Applying knowledge and understanding
Ability to convert various problems of practical interest into statistical models and make inference on it.

Making judgements
Students will be able to discern the different aspects of statistical modeling.

Communication skills
Students will properly use statistical and probabilistic language arising from the classical statistics.

Learning skills
The skills acquired will give students the opportunity of improving and deepening their knowledge of the different aspects of statistical modeling.

Course delivery

The course is composed of 72 hours of class lectures. Main lectures are devoted to the theorerical aspects of statistical inference; exercises will be assigned during these lectures. Lecture devoted to exercises are included in the course.

Learning assessment methods

The exam consists of two parts: the first part is a formal discussion of one of the main topics of statistical inference (70%), e.g., properties of random samples, estimators and principles of data reduction, point estimation, hypothesis testing, regression models; the second part consists of an exercise (30%).

Program

Properties of random samples

• random samples and their distributions;
• functions of random samples;
• Hoeffding's and Bernstein's inequality;
• Efron-Stein inequality;
• generating random samples;
• the likelihood function and the formal likelihood principle;
• exponential families of distributions.

Estimators and principles of data reduction

• sufficient statistics;
• minimal sufficient statistics;
• Fisher factorization and Lehmann-Scheffé theorem;
• finite-sample properties of estimators;
• Cramer-Rao lower bound and Rao-Blackwell theorem;
• large-sample properties of estimators.

Point estimation

• moment-based estimators;
• maximum likelihood estimators;
• the expectation-maximization algorithm;
• finite-sample properties of maximum likelihood estimators;
• large-sample properties of maximum likelihood estimators;
• Cramer theorem.

Hypothesis testing

• probabilistic structure of hypothesis tests;
• Neyman-Pearson lemma;
• likelihood ratio test;
• the Karlin-Rubin test;
• asymptotics for likelihood ration test;
• other large-sample hypothesis tests;
• hypothesis testing under the Gaussian model;
• oneway analysis of variance.

Regression models

• simple and multiple linear regression;
• least squares estimators and maximum likelihood estimators;
• Gauss-Markov theorem;
• hypothesis testing for regression models;
• generalized linear regression;
• logistic regression model;
• poisson regression models.

Bickel, P.J. and Doksum, K.A. (2015). Mathematical Statistics: basic ideas and selected topics. Chapman and Hall/CRC

Casella, G. and Berger, R.L. (2008). Statistical inference. Duxbury Press

Class schedule

• Teaching materials
• Exams and finals
• Go to Moodle
• Visit the forums