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. The student is introduced to statistical softwares R and WinBugs with particular reference to applications in finance and insurance.

Course contents

PART I: Classical Statistical Inference

Introduction.

Sampling distributions

Basic concepts of random samples. The likelihood function. Sample mean, sample variance. Central Limit Theorem. Sampling for the Gaussian distribution.

Estimation theory.

Point estimation.

Methods of finding point estimators: method of moments, least squares and maximum likelihood. Properties of point estimators: unbiasedness, efficiency, minimum variance. Mean squared error, Rao-Cramer inequality. Asymptotic properties: asymptotic unbiasedness, consistency. Properties of maximum likelihood estimators

 

Interval estimation.

Methods of finding interval estimators: frequentist interpretation of the confidence level. Pivotal quantities. Confidence interval for the mean of a Gaussian population. Asymptotic confidence interval for the mean of a non-Gaussian population. Confidence interval for the variance of a Gaussian population.

Hypotheses testing

Introduction to hypotheses testing: null hypothesis, alternative hypothesis, type I and II errors, power. Hypotheses testing in the Neyman-Pearson framework. Rejection and acceptance regions, p-value. Hypotheses testing for the mean and 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.
Introduction to WinBugs.
Case studies in finance and insurance.

Readings/Bibliography

Piccolo D. Statistica. Il Mulino, 2010

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

Teaching methods

Lab and 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.

Teaching tools

Software R and WinBugs

Office hours

See the website of Fedele Pasquale Greco

See the website of Carlo Trivisano