ENSAE Paris - École d'ingénieurs pour l'économie, la data science, la finance et l'actuariat
ENSAE Paris - École d'ingénieurs pour l'économie, la data science, la finance et l'actuariat
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Statistics 1

Teacher

Arnak DALALYAN

DALALYAN Arnak

Department: Statistics

ECTS:
4

Course Hours:
19.5

Tutorials Hours:
16.5

Language:
French

Examination Modality:
écrit+CC

Objective

This course provides the mathematical foundations of statistical modeling, primarily within a parametric framework. The inferential approach is developed with a focus on parameter estimation methods. The properties of estimators, particularly in terms of optimality, will be studied.

Learning outcomes: At the end of the course, the student will be able to:

  • Identify the nature of a statistical model: regular, discrete, density-based, identifiable;
  • Apply general parametric estimation methods, such as the method of moments estimator, the maximum likelihood estimator, and the Bayesian estimator;
  • Analyze the asymptotic behavior of estimators;
  • Assess the optimality of parametric estimation methods;
  • State and apply the general principles for constructing confidence regions and use them in common scenarios.

Assessment:

  • 2/3 of the final grade will be based on the written end-of-semester exam;
  • 1/3 of the grade will be based on continuous assessment, itself composed of a mid-term exam (50%), an attendance grade (25%), and a participation grade (25%).

Planning

Numbers in brackets correspond to the sections of the lecture notes.

  • Session 1: Limit theorems [1.4] + Continuity theorems [1.5]
  • Session 2: Sample [4.1], empirical distribution function [4.2], Glivenko-Cantelli
  • Session 3: Substitution method [4.3], statistics Xn-bar and S^2 [4.5 without Proposition 4.4]
  • Session 4: Identifiable statistical model, density-based, discrete [5.1] + consistency [5.2]
  • Session 5: Risk [5.2.1] + comparison [5.2.2] + Cramér-Rao inequality
  • Session 6: van Trees inequality and examples
  • Session 7: Generalized method of moments [5.3]. Likelihood and log-likelihood
  • Session 8: MLE: behavior of the log-likelihood [5.5] and consistency [5.6]
  • Session 9: MLE: regularity conditions and asymptotic normality
  • Session 10: MLE: geometric interpretation, asymptotic efficiency, and examples
  • Sessions 11 and 12: Bayesian estimation
  • Session 13: Confidence sets

References

Lehmann E.L. et G. Casella (2003) Theory of point estimation, 2nd edition, Springer-Verlag [21 LEH 00 D]
Tsybakov A. (2006) Polycopié du cours de Statistique Appliquée, Université Pierre et Marie Curie. Available in PDF format on Pamplemousse.
Wasserman L. (2004) All of Statistics, Springer-Verlag [21 WAS 00 A]

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