29515 - Advanced Statistics

Learning outcomes

By the end of the course, the student is aware of the basic methods for statistical inference both from a Frequentist and Bayesian perspective. More precisely, the student is able to address problems concerning parameter estimation and hypothesis testing within both inferential paradigms.

Course contents

PART I: Classical Statistical Inference

Introduction.

Parametric models. The likelihood function.

Estimation theory

Sufficiency and likelihood principle. Properties of the maximum likelihood estimators.

Hypotheses testing

Introduction to hypotheses testing: null hypothesis, alternative hypothesis, type I and II errors, power. Uniformly most powerful test. 

Non-parametric test: Kolmogorov-Smirnov test; testing for independence in contingency tables.

Interval estimation.

Relationship between hypothesis testing and interval estimation. Asymptotic confidence interval for the mean of a non-Gaussian population. Confidence interval for the variance of a Gaussian population.

 

PART II: Bayesian Statistical Inference

Introduction to Bayesian inference: the likelihood principle; prior and posterior distributions.
Summarizing posterior information.
Inference about parameters of some standard univariate models.
Relevance of Sufficient Statistics in Bayesian Inference. Conjugate priors.
Non informative priors and and reference priors.
Improper priors. The Jeffrey's rule.
Interval estimation. Hypothesis testing.
Introduction to Bayesian computational methods. Markov chain Monte Carlo methods.
Loss functions and posterior expected loss.
Hierarchical models.
Case studies in finance and insurance.

Readings/Bibliography

Piccolo D. Statistica. Il Mulino, 2010.

Azzalini A. Inferenza Statistica. Una presentazione basata sul concetto di verosimiglianza. Springer, 2001.

Lee P.M., Bayesian Statistics: an Introduction, Arnold, 2004.

Teaching methods

Classroom lectures

Assessment methods

The final examination aims at evaluating the achievement of the following objectives:

Deep knowledge concerning theoretical topics covered during the lectures;

Ability to analyze real data;

Ability to use WinBugs for Bayesian model estimation.

The final test will consist of computer session and an oral test.

Office hours

See the website of Fedele Pasquale Greco

See the website of Luca Scrucca