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

Statistical inference

Statistical inference

Academic year 2023/2024

Sommario del corso

• Course objectives
• Results of learning outcomes
• Program
• Course delivery
• Learning assessment methods
• Suggested readings and bibliography
• Class schedule
• Teaching tools
• Course card

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.

Program

Properties of random samples

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

Estimators and principles of data reduction

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

Point estimation

• moment-based estimators;
• maximum likelihood estimators;
• expectation-maximization and Newton's type algorithms;
• finite-sample properties of maximum likelihood estimators;
• strong consistency of maximum likelihood estimators;
• asymptotic normality of maximum likelihood estimators;
• asymptotic efficiency.

Hypothesis testing

• probabilistic structure of hypothesis tests;
• Neyman-Pearson lemma;
• likelihood ratio test;
• monotone likelihood ration and Karlin-Rubin test;
• asymptotic distribution of likelihood ration tests;
• 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.

Course delivery

The course is composed of 72 hours of lectures, including lectures dedicated to excercices.

The course is composed of 72 hours of lectures, including lectures dedicated to excercices. Until further notice, for the AY 2023/2024 the teaching modality is foreseen to be in presence.

Learning assessment methods

The learning assessment method consists in a written test followed by an oral examination, both in English.

The written test consists in two exercises on the topics treated in the lectures, and it has a duration of 90 minutes. The mark will be expressed in thirtieth, with 30 points distributed between the two exercises on the basis of their difficulty. The mark will be given by summing up the points obtained in each exercise. 

The oral examination is scheduled the day after the written test, and it can be given only if the written test is passed, i.e. a mark of 18 or better. The oral examination consists in questions on the exercises of the written test, and on topics treated in the lectures. The mark will be expressed in thirtieth.

If the oral examination is passed, i.e. a mark of 18 or better, the final mark will consist of a combination between the mark obtained at the written test (50%) and the mark obtained at the oral examination (50%). The minimal final mark to pass the exam is 18.

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. Cengage Learning.

Keener R.W. (2010). Theoretical statistics: topics for a core course. Springer.

Lehmann, E.L. and Casella, G. (2003). Theory of point estimation. Springer.

Class schedule

• Teaching material
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