Introduction to Statistical Inference. Principles of data reductions. Point estimation. Testing of Hypotheses. Interval estimation. An introduction to some advanced topics. Monte Carlo statistical methods.
Casella, G. and Berger L. R. (2002). Statistical Inference. Second Edition. Duxbury Press.
The aim of this course is to introduce students to the theory and practice of statistical inference and to the related computational strategies and algorithms.
Moreover the course aims to develop
students' expertise to analyze inferential problems selecting the appropriate inferential methods, to implement the analyses in R, and to properly interpreter and describe the results of the analyses.
Basic knowledge of algebra, maths and statistics
Lectures, sessions of exercises and labs
Additional teaching material will be provided during the course through the e-learning platform
Type of Assessment
The exam consists of a written test and an oral exam, aimed to evaluate students' comprehension, acquisition and laboration skills. The written test includes both applied exercises (that may require the use of a pocket calculator) and theoretical exercises. Students must pass the written test to take the oral exam. The oral exam includes theoretical questions and a discussion of a project of data analysis with R.
INTRODUCTION TO STATISTICAL INFERENCE. Statistical model. Parametric statistical model. Exponential families. Regular exponential families. Location and scale families. Experiment and statistical model. Sampling from infinite population. Random sample and observed sample. Statistical models for random samples. An introduction to the main inferential problems. Inferential approaches: an overview.
STATISTICS AND SAMPLING DISTRIBUTIONS. Statistics. Sampling distributions. Sum, mean and variance for random samples. Sufficient statistics. Ancillary statistics. Complete statistics. Basu's theorem.
POINT ESTIMATION. Estimators: minimax estimators, Method of moments estimators, maximum Likelihood estimators. Properties of estimators: Unbiasedness, efficiency. Fisher information. Cramér-Rao inequality. Rao-Blackwell's theorem. Lehmann-Scheffè theorem. Asymptotic theory: consistency, estimators with Normal asymptotic distributions, asymptotic efficiency. Properties of Likelihood estimators. Properties of method of moments estimators.
TESTING OF HYPOTHESES. Formulation of the hypothesis testing problem. The Neyman-Pearson Theorem. Likelihood ratio tests. Most powerful tests.
Unbiased test. Asymptotic tests. Important examples. P-value: definition and interpretation. Some thought on the concepts of p-value and statistical significance. Introduction to multiple test.
INTERVAL ESTIMATION, Pivotal quantities and method of pivots. Method of test inversion. Methods of evaluating interval estimators. Approximate maximum likelihood intervals.
MISCELLANEA. The Delta method. Bootstrap. The Newton-Raphson algorithm. The EM algorithm. Introduction to mixture models. Introduction to nonparametric Statistical Inference.
MONTE CARLO STATISTICAL METHODS. Deterministic Numerical Methods: Optimization and Integration. Random Variable Generation: Uniform simulation; The inverse transform; General transformation methods (A normal generator; Discrete distributions; Mixture representations); Accept–reject methods. Monte Carlo Integration: Classical Monte Carlo integration; Importance sampling. Controlling and Accelerating Convergence. Monte Carlo Optimization: Numerical optimization methods; Stochastic approximation (Optimizing Monte Carlo approximations; The EM algorithm; Monte Carlo EM).