# Fundamentals of statistical inference

Pier Giovanni Bissiri, Maria-Pia Victoria Feser

• Date:

08 JANUARY
-
30 MARCH 2024

• Event location: Aula IV, Department of Statistical Sciences, Via Belle Arti 41 (STAT PhD Classes Virtual Room on need)

• Type: Cycle 39 - Mandatory Courses

Aims: to explore a number of statistical principles. To consider the nature of statistical parameters, the different viewpoints of Bayesian and Frequentist approaches and their relationship with the given statistical principles. To introduce the idea of inference as a statistical decision problem. To introduce important aspects of statistical modelling, including model selection, various extensions to likelihood theory and its link with alternative approaches, to master asymptotic arguments.

Learning outcomes: An appreciation for the complexity of statistical inference, recognition of its inherent subjectivity and the role of expert judgement, the ability to critique familiar inference methods, knowledge of the key choices that must be made, and scepticism about apparently simple answers to difficult questions.

Final exam: written exam for each modulus.

Course contents
The course is articulated in two moduli, with final assessment for each of the moduli.

Module 1: Foundations of Statistics (prof. P.G.Bissiri)
- Statistics, Sufficiency, and Completeness
- Efficiency of estimators, optimality of tests
- The Bayesian approach, exchangeability, Dirichlet process

Module 2: Beyond conventional statistical inference (prof. M.V. Feser)

1) Assessing Estimators and Testing Procedures
- Consistency and Finite Sample Bias
- Finite Sample Distribution
- Finite Sample Size and Coverage
- The Influence Function and the Sensitivity Curve
- The Breakdown Point

2) Robust Estimators
- M-Estimators
- Bounded IF M-Estimators
- The choice of the bound c
- The linear model

3) Alternative Robust Estimators
- S-Estimators
- Weighted MLE
- Numerical Methods
- Generalized Linear Models

References

Module 1:
Shao, J. (2003). Mathematical Statistics. Springer
Schervish, Mark J. (1995). Theory of Statistics. Springer

Module 2:
Heritier, S., E. Cantoni, S. Copt and M.-P. Victoria-Feser, Robust Methods in Biostatistics, 2009, Wiley Series in Probability and Statistics. DOI:10.1002/9780470740538.
Efron, B. and R.J. Tibshirani, 1994, An Introduction to the Bootstrap, Chapman & Hall/CRC Monographs on Statistics and Applied Probability.https://doi.org/10.1201/9780429246593