Advanced Topics in Statistical Inference

General characterization

Prerequisites

Basic knowledge of calculus and linear algebra, such as topologic notions, integral and differential calculus, functions of more then one variable and matricial calculus.  

Bibliography

Casella G e Berger RL (2002). Statistical Inference - 2nd edition. Duxbury Press.

Paulino CD, Amaral-Turkman MA, Murteira B, Silva GL (2018). Estatística Bayesiana. Fundação Calouste Gulbenkian.

Mood, Graybill e Boes (1974). Introduction to the Theory of Statistics. McGraw-Hill.

Rohatgi VK, Saleh AK (2001). An Introduction to Probability Theory and Mathematical Statistics. Wiley & Sons.

Box GEP, Tiao GC (1973). Bayesian Inference in Statistical Analysis. Wiley-Interscience.

DeGroot MH (1989). Probability and Statistics - 2nd edition. Addison-Wesley.

Paulino CD (2000). Notas de Inferência Estatística. Associação de Estudantes do IST.

Teaching method

Classes/Labs with the students participation, where the concepts are explained and problems solved.

Evaluation method

Four theoretical-practical assignments (25% each).

Subject matter

Preliminary concepts of Statistical Inference
-Common families of distributions - Exponential Family, Localization and Scale Families
-Random Samples and their Properties
- Basic Concepts, Sampling from the Normal Distribution, Ordinal Statistics, Convergence Concepts (Convergency in Probability, All Most Sure Convergency and Convergency in Distribution, Strong Law of the Large Numbers, Central Limit Theorem, Slutsky Theorem, Delta Method)

Data Reduction
-Sufficient and Minimal Sufficient Statistics, Factorization Criteria, Ancillary and Complete Statistics, Relationship between the several types of Statistics

Parametric Point Estimation
-Properties of the Estimators (Mean Square Error, Unbiased Estimator, Relative Efficiency)
- Optimality Criteria of Estimators (Best Linear Unbiased Estimation, Unbiased Estimation with Uniformly Minimum Variance, More Efficient Estimation, Cramer-Rao Inferior Limit, Fisher Information, Sufficiency and Unbiased Estimation)
- Estimation Methods (Method of Moments, Method of Maximum Likelihood, Least Square Method, Others)

Hypothesis testing
-Hypothesis Tests and Criteria for Evaluating the Tests (Definition of the Hypothesis, Test Statistcs, Probabilities of Type I and Type II Errors, Power Function, Size and Level of a Test, Unbiased Tests, More Powerful Tests, Uniformly More Powerful Tests, Families of Monotonic Likelihood Ratio)
-Methods for Constructing Tests (Likelihood Ratio Tests, Using Optimality Criteria)

Interval Parametric Estimation
-Methods for Constructing Interval Estimators (Duality with Hypothesis Tests, Pivotal Method)
-Optimality Criteria for Interval Estimators (Size and Coverage Probability,Optimality of the Related Test)

Analysis of Variance and Regression
-Simple Regression Analysis (Models and Distributional Assumptions, Estimation and Tests Assuming Normal Errors, Estimation and Prediction)
-Simple Analysis of Variance (Models and Distributional Assumptions, The Classic ANOVA Hypothesis, Inference for Linear Combinations of the Means, The ANOVA F-Test, Simultaneous Estimation of the Contrasts, Partition of the Square Sums)

Introduction to Bayesian Statistics
-Introduction to Bayesian Statistics
-Bayes Theorem. Bayesian Paradigm
-Lihelihood Model
-Prior Distribution. Non Informative and Conjugate Priors
-Bayesian Inference

Programs

Programs where the course is taught: