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# Cours scientifiques - APM_3F004_EP : Asymptotic Statistics

## Descriptif

The course MAA304 begins with a detailed overview of convergence, both in probability and in distribution, and revisiting two key theorems in statistics: the law of large numbers and the central limit theorem. We will then look in detail at asymptotic statistics, including fundamental topics such as the asymptotic properties of maximum likelihood estimators (MLEs), the formulation of asymptotic confidence intervals, and the principles underlying asymptotic test theory.

We will then highlight the crucial role of information theory in statistics, with particular emphasis on notions of efficiency, Cramer-Rao theory, and sufficiency. Moving to multivariate linear regression, the focus shifts to inference in Gaussian models and model validation to give students a solid understanding of this important statistical paradigm.

Next, we turn to nonlinear regression and delve into a comprehensive study of logistic regression. The course concludes with a brief introduction to nonparametric statistics, emphasising the importance of distribution-free tests.

## Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique

Vous devez avoir validé l'équation suivante : UE FMA_2F002_EP Et UE APM_2F010_EP

Numérique sur 20

## Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique

### Vos modalités d'acquisition :

Midterm Exam - 2hours, closed book, written exam

Final Exam - 3 hours, closed book, written exam

Assignments and Quizzes: 5 group assignements [homework] and 2 quizzes [1/2h, written]

Le rattrapage est autorisé (Note de rattrapage conservée écrêtée à une note seuil de 10)
L'UE est acquise si Note finale >= 9
• Crédits ECTS acquis : 5 ECTS

La note obtenue rentre dans le calcul de votre GPA.

### Programme détaillé

1. Introduction to Asymptotic Statistics: Define and explain the key concepts, history, and importance of asymptotic statistics.

2. Convergence Concepts: Discuss convergence in probability, almost sure convergence, convergence in distribution, and convergence in quadratic mean.

3. Overview of Probability Theory: Explore the underlying probability theory necessary for understanding asymptotic statistics.

4. Law of Large Numbers: Discuss the law of large numbers, its proof, and applications in statistics.

5. Central Limit Theorem: Explain the central limit theorem, its proof, and its implications for statistical analysis.

6. Asymptotic Properties of Maximum Likelihood Estimators (MLE): Explain asymptotic normality and efficiency and give examples.

7. Asymptotic Confidence Intervals: Derive asymptotic confidence intervals and compare them to standard intervals.

8. Asymptotic Test Theory: Discuss basic asymptotic test theory, including likelihood ratio, Wald, and score tests.

9. Cramer-Rao Theory: Discuss the Cramer-Rao bound and the efficiency of estimators.

10. Summary and Applications: Recap key concepts and discuss practical applications of asymptotic statistics in various fields.

11  Introduction to Linear Regression: Cover the basics of linear regression, its assumptions, the meaning of coefficients, and real-world applications.

12  Simple Linear Regression: Discuss simple linear regression with one predictor variable, including model fitting, interpretation, and assumptions checking.

13  Multiple Linear Regression: Explore multiple linear regression, including the handling of categorical predictors and interaction terms.

14  Model Selection and Validation: Learn about strategies for model selection, including methods like forward selection, backward elimination, and cross-validation. Discuss model diagnostics and how to address issues like multicollinearity and heteroscedasticity.

15  Advanced Topics and Applications: Examine the use of linear regression in more complex scenarios, such as polynomial and logistic regression, regularization techniques (Ridge, Lasso), and its application in machine learning.

16 Understanding Logistic Regression: Begin with an introduction to logistic regression, explaining its role in dealing with categorical dependent variables, especially binary outcomes, and its applications in various fields.

17 Modeling with Logistic Regression: Discuss the basics of logistic regression modeling, explaining concepts like odds ratios, logit function, and how to interpret the results of a logistic regression analysis.

18 Introduction to Nonparametric Distribution-Free Tests: Introduce the concept of nonparametric tests, emphasizing their use when data doesn't conform to a specific distribution or assumptions of parametric tests are violated.

19 Common Distribution-Free Tests: Discuss common nonparametric tests, such as the Mann-Whitney U test, Kruskal-Wallis test, and the Kolmogorov-Smirnov test, explaining their purpose, application, and interpretation.