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Department of Mathematics "Giuseppe Peano"

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

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## Statistical inference

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Course ID
MAT0035
Teacher
Stefano Favaro
Year
1st year
Teaching period
First semester
Type
D.M. 270 TAF C - Related or integrative
Credits/Recognition
9
Course disciplinary sector (SSD)
SECS-S/01 - statistica
Delivery
Formal authority
Language
English
Attendance
Mandatory
Type of examination
Written
Prerequisites
Mathematical, probabilistic and statistical tools acquired in the three-year undergraduate program. A detailed list of the required backgroud will be provided during the first lecture.
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## Course objectives

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

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## 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.

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## 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.
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## 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.

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## Learning assessment methods

During the Covid-19 emergency the learning assessment method consists in a written exam with video surveillance on Webex.

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%).

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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.

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## Class schedule Enroll
• Open
Enrollment opening date
01/09/2019 at 00:00
Enrollment closing date
30/06/2020 at 00:00
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Last update: 14/05/2020 09:24
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