# Coursebooks

## Statistical theory

#### Lecturer(s) :

Koch Erwan Fabrice

English

#### Summary

The course aims at developing certain key aspects of the theory of statistics, providing a common general framework for statistical methodology. While the main emphasis will be on the mathematical aspects of statistics, an effort will be made to balance rigor and intuition.

#### Content

• Stochastic convergence and its use in statistics: modes of convergence, weak law of large numbers, central limit theorem.
• Formalization of a statistical problem : parameters, models, parametrizations, sufficiency, ancillarity, completeness.
• Point estimation: methods of estimation, bias, variance, relative efficiency.
• Likelihood theory: the likelihood principle, asymptotic properties, misspecification of models, the Bayesian perspective.
• Optimality: decision theory, minimum variance unbiased estimation, Cramér-Rao lower bound, efficiency, robustness.
• Testing and Confidence Regions: Neyman-Pearson setup, likelihood ratio tests, uniformly most powerful (UMP) tests, duality with confidence intervals, confidence regions, large sample theory, goodness-of-fit testing.

#### Learning Prerequisites

##### Recommended courses

Real Analysis, Linear Algebra, Probability, Statistics.

#### Learning Outcomes

By the end of the course, the student must be able to:
• Formulate the various elements of a statistical problem rigorously.
• Formalize the performance of statistical procedures through probability theory.
• Systematize broad classes of probability models and their structural relation to inference.
• Construct efficient statistical procedures for point/interval estimation and testing in classical contexts.
• Derive certain exact (finite sample) properties of fundamental statistical procedures.
• Derive certain asymptotic (large sample) properties of fundamental statistical procedures.
• Formulate fundamental limitations and uncertainty principles of statistical theory.
• Prove certain fundamental structural and optimality theorems of statistics.

#### Teaching methods

Lecture ex cathedra using slides as well as the blackboard (especially for some proofs). Examples/exercises presented/solved at the blackboard.

#### Assessment methods

Final written exam.

Dans le cadre de l¿art. 3 al. 5 du Règlement de section, l¿enseignant décide de la forme de l¿examen qu¿il communique aux étudiants concernés.

#### Supervision

 Office hours No Assistants Yes Forum Yes

#### Resources

##### Notes/Handbook

The slides will be available on Moodle.

### In the programs

• Mathematics - master program, 2019-2020, Master semester 1
• Semester
Fall
• Exam form
Written
• Credits
5
• Subject examined
Statistical theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Mathematics - master program, 2019-2020, Master semester 3
• Semester
Fall
• Exam form
Written
• Credits
5
• Subject examined
Statistical theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Applied Mathematics, 2019-2020, Master semester 1
• Semester
Fall
• Exam form
Written
• Credits
5
• Subject examined
Statistical theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Applied Mathematics, 2019-2020, Master semester 3
• Semester
Fall
• Exam form
Written
• Credits
5
• Subject examined
Statistical theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Data Science, 2019-2020, Master semester 1
• Semester
Fall
• Exam form
Written
• Credits
5
• Subject examined
Statistical theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Data Science, 2019-2020, Master semester 3
• Semester
Fall
• Exam form
Written
• Credits
5
• Subject examined
Statistical theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks

MoTuWeThFr
8-9 MAA112
9-10
10-11 MAA112
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
Lecture
Exercise, TP
Project, other

### legend

• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German